Hyperbolic sine

Percentage Accurate: 54.5% → 100.0%

Time: 9.2s

Alternatives: 9

Speedup: 29.4×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \frac{e^{x} - e^{-x}}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- (exp x) (exp (- x))) 2.0))

C

double code(double x) {
	return (exp(x) - exp(-x)) / 2.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (exp(x) - exp(-x)) / 2.0d0
end function

Java

public static double code(double x) {
	return (Math.exp(x) - Math.exp(-x)) / 2.0;
}

Python

def code(x):
	return (math.exp(x) - math.exp(-x)) / 2.0

Julia

function code(x)
	return Float64(Float64(exp(x) - exp(Float64(-x))) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (exp(x) - exp(-x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Initial Program: 54.5% 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

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.005:\\ \;\;\;\;\frac{\mathsf{fma}\left(x\_m \cdot x\_m, 0.016666666666666666, 0.3333333333333333\right) \cdot {x\_m}^{3} + x\_m \cdot 2}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #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 0.005)
      (/
       (+
        (*
         (fma (* x_m x_m) 0.016666666666666666 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 <= 0.005) {
		tmp = ((fma((x_m * x_m), 0.016666666666666666, 0.3333333333333333) * 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 <= 0.005)
		tmp = Float64(Float64(Float64(fma(Float64(x_m * x_m), 0.016666666666666666, 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

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, 0.005], N[(N[(N[(N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.016666666666666666 + 0.3333333333333333), $MachinePrecision] * 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))) < 0.0050000000000000001

   1. Initial program 36.1%

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

   3. Taylor expanded in x around 0 93.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {x}^{2}\right)\right)}}{2} \]
   4. Step-by-step derivation

      1. *-commutative 93.9%

         \[\leadsto \frac{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{x}^{2} \cdot 0.016666666666666666}\right)\right)}{2} \]

   5. Simplified 93.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot 0.016666666666666666\right)\right)}}{2} \]
   6. Step-by-step derivation

      1. +-commutative 93.9%

         \[\leadsto \frac{x \cdot \color{blue}{\left({x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot 0.016666666666666666\right) + 2\right)}}{2} \]

      2. distribute-lft-in 93.9%

         \[\leadsto \frac{\color{blue}{x \cdot \left({x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot 0.016666666666666666\right)\right) + x \cdot 2}}{2} \]

      3. *-commutative 93.9%

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot 0.016666666666666666\right)\right) \cdot x} + x \cdot 2}{2} \]

      4. *-commutative 93.9%

         \[\leadsto \frac{\color{blue}{\left(\left(0.3333333333333333 + {x}^{2} \cdot 0.016666666666666666\right) \cdot {x}^{2}\right)} \cdot x + x \cdot 2}{2} \]

      5. associate-*l* 93.9%

         \[\leadsto \frac{\color{blue}{\left(0.3333333333333333 + {x}^{2} \cdot 0.016666666666666666\right) \cdot \left({x}^{2} \cdot x\right)} + x \cdot 2}{2} \]

      6. +-commutative 93.9%

         \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot 0.016666666666666666 + 0.3333333333333333\right)} \cdot \left({x}^{2} \cdot x\right) + x \cdot 2}{2} \]

      7. fma-define 93.9%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, 0.016666666666666666, 0.3333333333333333\right)} \cdot \left({x}^{2} \cdot x\right) + x \cdot 2}{2} \]

      8. unpow2 93.9%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{2}, 0.016666666666666666, 0.3333333333333333\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + x \cdot 2}{2} \]

      9. pow3 93.9%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{2}, 0.016666666666666666, 0.3333333333333333\right) \cdot \color{blue}{{x}^{3}} + x \cdot 2}{2} \]

   7. Applied egg-rr 93.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{2}, 0.016666666666666666, 0.3333333333333333\right) \cdot {x}^{3} + x \cdot 2}}{2} \]
   8. Step-by-step derivation

      1. unpow2 93.9%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.016666666666666666, 0.3333333333333333\right) \cdot {x}^{3} + x \cdot 2}{2} \]

   9. Applied egg-rr 93.9%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.016666666666666666, 0.3333333333333333\right) \cdot {x}^{3} + x \cdot 2}{2} \]

   if 0.0050000000000000001 < (-.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: 100.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_0 := e^{x\_m} - e^{-x\_m}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_0 \leq 0.005:\\ \;\;\;\;\frac{x\_m \cdot \left(2 + {x\_m}^{2} \cdot \left(0.3333333333333333 + \left(x\_m \cdot x\_m\right) \cdot 0.016666666666666666\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_0}{2}\\ \end{array} \end{array} \end{array} \]

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 0.005)
      (/
       (*
        x_m
        (+
         2.0
         (*
          (pow x_m 2.0)
          (+ 0.3333333333333333 (* (* x_m x_m) 0.016666666666666666)))))
       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 <= 0.005) {
		tmp = (x_m * (2.0 + (pow(x_m, 2.0) * (0.3333333333333333 + ((x_m * x_m) * 0.016666666666666666))))) / 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 <= 0.005d0) then
        tmp = (x_m * (2.0d0 + ((x_m ** 2.0d0) * (0.3333333333333333d0 + ((x_m * x_m) * 0.016666666666666666d0))))) / 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 <= 0.005) {
		tmp = (x_m * (2.0 + (Math.pow(x_m, 2.0) * (0.3333333333333333 + ((x_m * x_m) * 0.016666666666666666))))) / 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 <= 0.005:
		tmp = (x_m * (2.0 + (math.pow(x_m, 2.0) * (0.3333333333333333 + ((x_m * x_m) * 0.016666666666666666))))) / 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 <= 0.005)
		tmp = Float64(Float64(x_m * Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.3333333333333333 + Float64(Float64(x_m * x_m) * 0.016666666666666666))))) / 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 <= 0.005)
		tmp = (x_m * (2.0 + ((x_m ^ 2.0) * (0.3333333333333333 + ((x_m * x_m) * 0.016666666666666666))))) / 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, 0.005], N[(N[(x$95$m * N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $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))) < 0.0050000000000000001

   1. Initial program 36.1%

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

   3. Taylor expanded in x around 0 93.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {x}^{2}\right)\right)}}{2} \]
   4. Step-by-step derivation

      1. *-commutative 93.9%

         \[\leadsto \frac{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{x}^{2} \cdot 0.016666666666666666}\right)\right)}{2} \]

   5. Simplified 93.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot 0.016666666666666666\right)\right)}}{2} \]
   6. Step-by-step derivation

      1. unpow2 93.9%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.016666666666666666, 0.3333333333333333\right) \cdot {x}^{3} + x \cdot 2}{2} \]

   7. Applied egg-rr 93.9%

      \[\leadsto \frac{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.016666666666666666\right)\right)}{2} \]

   if 0.0050000000000000001 < (-.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 3: 99.7% accurate, 1.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3:\\ \;\;\;\;\frac{x\_m \cdot \left(2 + {x\_m}^{2} \cdot \left(0.3333333333333333 + \left(x\_m \cdot x\_m\right) \cdot 0.016666666666666666\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x\_m} + -1}{2}\\ \end{array} \end{array} \]

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 3.0)
    (/
     (*
      x_m
      (+
       2.0
       (*
        (pow x_m 2.0)
        (+ 0.3333333333333333 (* (* x_m x_m) 0.016666666666666666)))))
     2.0)
    (/ (+ (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 <= 3.0) {
		tmp = (x_m * (2.0 + (pow(x_m, 2.0) * (0.3333333333333333 + ((x_m * x_m) * 0.016666666666666666))))) / 2.0;
	} 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 <= 3.0d0) then
        tmp = (x_m * (2.0d0 + ((x_m ** 2.0d0) * (0.3333333333333333d0 + ((x_m * x_m) * 0.016666666666666666d0))))) / 2.0d0
    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 <= 3.0) {
		tmp = (x_m * (2.0 + (Math.pow(x_m, 2.0) * (0.3333333333333333 + ((x_m * x_m) * 0.016666666666666666))))) / 2.0;
	} 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 <= 3.0:
		tmp = (x_m * (2.0 + (math.pow(x_m, 2.0) * (0.3333333333333333 + ((x_m * x_m) * 0.016666666666666666))))) / 2.0
	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 <= 3.0)
		tmp = Float64(Float64(x_m * Float64(2.0 + Float64((x_m ^ 2.0) * Float64(0.3333333333333333 + Float64(Float64(x_m * x_m) * 0.016666666666666666))))) / 2.0);
	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 <= 3.0)
		tmp = (x_m * (2.0 + ((x_m ^ 2.0) * (0.3333333333333333 + ((x_m * x_m) * 0.016666666666666666))))) / 2.0;
	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, 3.0], N[(N[(x$95$m * N[(2.0 + N[(N[Power[x$95$m, 2.0], $MachinePrecision] * N[(0.3333333333333333 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.016666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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 < 3

   1. Initial program 36.1%

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

   3. Taylor expanded in x around 0 93.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + 0.016666666666666666 \cdot {x}^{2}\right)\right)}}{2} \]
   4. Step-by-step derivation

      1. *-commutative 93.9%

         \[\leadsto \frac{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{{x}^{2} \cdot 0.016666666666666666}\right)\right)}{2} \]

   5. Simplified 93.9%

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + {x}^{2} \cdot 0.016666666666666666\right)\right)}}{2} \]
   6. Step-by-step derivation

      1. unpow2 93.9%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.016666666666666666, 0.3333333333333333\right) \cdot {x}^{3} + x \cdot 2}{2} \]

   7. Applied egg-rr 93.9%

      \[\leadsto \frac{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + \color{blue}{\left(x \cdot x\right)} \cdot 0.016666666666666666\right)\right)}{2} \]

   if 3 < 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 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3:\\ \;\;\;\;\frac{x \cdot \left(2 + {x}^{2} \cdot \left(0.3333333333333333 + \left(x \cdot x\right) \cdot 0.016666666666666666\right)\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} + -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2.15:\\ \;\;\;\;\frac{x\_m \cdot \left(2 + \left(x\_m \cdot x\_m\right) \cdot 0.3333333333333333\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x\_m} + -1}{2}\\ \end{array} \end{array} \]

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 2.15)
    (/ (* x_m (+ 2.0 (* (* x_m x_m) 0.3333333333333333))) 2.0)
    (/ (+ (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 <= 2.15) {
		tmp = (x_m * (2.0 + ((x_m * x_m) * 0.3333333333333333))) / 2.0;
	} 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 <= 2.15d0) then
        tmp = (x_m * (2.0d0 + ((x_m * x_m) * 0.3333333333333333d0))) / 2.0d0
    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 <= 2.15) {
		tmp = (x_m * (2.0 + ((x_m * x_m) * 0.3333333333333333))) / 2.0;
	} 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 <= 2.15:
		tmp = (x_m * (2.0 + ((x_m * x_m) * 0.3333333333333333))) / 2.0
	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 <= 2.15)
		tmp = Float64(Float64(x_m * Float64(2.0 + Float64(Float64(x_m * x_m) * 0.3333333333333333))) / 2.0);
	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 <= 2.15)
		tmp = (x_m * (2.0 + ((x_m * x_m) * 0.3333333333333333))) / 2.0;
	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, 2.15], N[(N[(x$95$m * N[(2.0 + N[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $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 < 2.14999999999999991

   1. Initial program 36.1%

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

   3. Taylor expanded in x around 0 92.4%

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

      1. unpow2 93.9%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.016666666666666666, 0.3333333333333333\right) \cdot {x}^{3} + x \cdot 2}{2} \]

   5. Applied egg-rr 92.4%

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

   if 2.14999999999999991 < 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 94.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.15:\\ \;\;\;\;\frac{x \cdot \left(2 + \left(x \cdot x\right) \cdot 0.3333333333333333\right)}{2}\\ \mathbf{else}:\\ \;\;\;\;\frac{e^{x} + -1}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 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 49.6%

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

3. Taylor expanded in x around 0 25.6%

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

   1. mul-1-neg 25.6%

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

   2. unsub-neg 25.6%

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

5. Simplified 25.6%

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

6. Taylor expanded in x around 0 70.1%

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

   1. *-commutative 70.1%

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

8. Simplified 70.1%

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

Alternative 6: 84.4% accurate, 18.7× speedup

Math

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

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 (* (* x_m x_m) 0.3333333333333333))) 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 + ((x_m * x_m) * 0.3333333333333333))) / 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 + ((x_m * x_m) * 0.3333333333333333d0))) / 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 + ((x_m * x_m) * 0.3333333333333333))) / 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 + ((x_m * x_m) * 0.3333333333333333))) / 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(Float64(x_m * x_m) * 0.3333333333333333))) / 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 + ((x_m * x_m) * 0.3333333333333333))) / 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[(N[(x$95$m * x$95$m), $MachinePrecision] * 0.3333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 49.6%

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

3. Taylor expanded in x around 0 88.2%

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

   1. unpow2 92.3%

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{x \cdot x}, 0.016666666666666666, 0.3333333333333333\right) \cdot {x}^{3} + x \cdot 2}{2} \]

5. Applied egg-rr 88.2%

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

6. Final simplification 88.2%

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

Alternative 7: 75.2% 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 49.6%

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

3. Taylor expanded in x around 0 25.6%

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

   1. mul-1-neg 25.6%

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

   2. unsub-neg 25.6%

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

5. Simplified 25.6%

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

6. Taylor expanded in x around 0 65.3%

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

   1. *-commutative 65.3%

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

8. Simplified 65.3%

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

Alternative 8: 52.1% 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 49.6%

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

3. Taylor expanded in x around 0 88.2%

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

4. Taylor expanded in x around 0 56.2%

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

Alternative 9: 4.3% accurate, 206.0× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot 13 \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m) :precision binary64 (* x_s 13.0))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m) {
	return x_s * 13.0;
}

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 * 13.0d0
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 * 13.0;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m):
	return x_s * 13.0

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m)
	return Float64(x_s * 13.0)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m)
	tmp = x_s * 13.0;
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 * 13.0), $MachinePrecision]

Derivation

1. Initial program 49.6%

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

4. Applied egg-rr 27.4%

   \[\leadsto \frac{\color{blue}{27} - e^{-x}}{2} \]

5. Taylor expanded in x around 0 2.8%

   \[\leadsto \color{blue}{13} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024114 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))