Hyperbolic sine

Percentage Accurate: 53.8% → 100.0%

Time: 7.7s

Alternatives: 7

Speedup: 12.8×

• 49.1% 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: 53.8% 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, 2.1× speedup

Math

\[\begin{array}{l} \\ \sinh x \end{array} \]

FPCore

(FPCore (x) :precision binary64 (sinh x))

C

double code(double x) {
	return sinh(x);
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = sinh(x)
end function

Java

public static double code(double x) {
	return Math.sinh(x);
}

Python

def code(x):
	return math.sinh(x)

Julia

function code(x)
	return sinh(x)
end

MATLAB

function tmp = code(x)
	tmp = sinh(x);
end

Wolfram

code[x_] := N[Sinh[x], $MachinePrecision]

Derivation

1. Initial program 52.7%

   \[\frac{e^{x} - e^{-x}}{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sinh-undef N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sinh.f64 100.0

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

4. Applied egg-rr 100.0%

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

   1. lift-sinh.f64 N/A

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

   2. associate-/l* N/A

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

   3. metadata-eval N/A

      \[\leadsto \sinh x \cdot \color{blue}{1} \]

   4. *-rgt-identity 100.0

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

6. Applied egg-rr 100.0%

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

Alternative 2: 67.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 0.002:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 0.002)
   x
   (* 0.16666666666666666 (* x (* x x)))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 0.002) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (x * x));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if ((exp(x) - exp(-x)) <= 0.002d0) then
        tmp = x
    else
        tmp = 0.16666666666666666d0 * (x * (x * x))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if ((Math.exp(x) - Math.exp(-x)) <= 0.002) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (x * x));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if (math.exp(x) - math.exp(-x)) <= 0.002:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (x * (x * x))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 0.002)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(x * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if ((exp(x) - exp(-x)) <= 0.002)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (x * (x * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 0.002], x, N[(0.16666666666666666 * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 2e-3

   1. Initial program 37.5%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 69.2

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

   5. Simplified 69.2%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. metadata-eval N/A

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

      4. *-rgt-identity 69.2

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

   7. Applied egg-rr 69.2%

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

   if 2e-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. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + \frac{1}{3} \cdot {x}^{2}\right)}}{2} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. unpow2 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\frac{1}{3}, \color{blue}{x \cdot x}, 2\right)}{2} \]

      5. lower-*.f64 68.2

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

   5. Simplified 68.2%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 N/A

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

      2. cube-mult N/A

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

      3. unpow2 N/A

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

      4. lower-*.f64 N/A

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

      5. unpow2 N/A

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

      6. lower-*.f64 68.2

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

   8. Simplified 68.2%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(x \cdot \left(x \cdot x\right)\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 75.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right)\\ \mathbf{if}\;x \leq 5 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{x \cdot \mathsf{fma}\left(x \cdot x, t\_0 \cdot t\_0, -4\right)}{\mathsf{fma}\left(x, t\_0, -2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (fma x (* x 0.016666666666666666) 0.3333333333333333))))
   (if (<= x 5e+60)
     (/ (/ (* x (fma (* x x) (* t_0 t_0) -4.0)) (fma x t_0 -2.0)) 2.0)
     (fma
      (fma (* x x) 0.008333333333333333 0.16666666666666666)
      (* x (* x x))
      x))))

C

double code(double x) {
	double t_0 = x * fma(x, (x * 0.016666666666666666), 0.3333333333333333);
	double tmp;
	if (x <= 5e+60) {
		tmp = ((x * fma((x * x), (t_0 * t_0), -4.0)) / fma(x, t_0, -2.0)) / 2.0;
	} else {
		tmp = fma(fma((x * x), 0.008333333333333333, 0.16666666666666666), (x * (x * x)), x);
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * fma(x, Float64(x * 0.016666666666666666), 0.3333333333333333))
	tmp = 0.0
	if (x <= 5e+60)
		tmp = Float64(Float64(Float64(x * fma(Float64(x * x), Float64(t_0 * t_0), -4.0)) / fma(x, t_0, -2.0)) / 2.0);
	else
		tmp = fma(fma(Float64(x * x), 0.008333333333333333, 0.16666666666666666), Float64(x * Float64(x * x)), x);
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * N[(x * 0.016666666666666666), $MachinePrecision] + 0.3333333333333333), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 5e+60], N[(N[(N[(x * N[(N[(x * x), $MachinePrecision] * N[(t$95$0 * t$95$0), $MachinePrecision] + -4.0), $MachinePrecision]), $MachinePrecision] / N[(x * t$95$0 + -2.0), $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision], N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 4.99999999999999975e60

   1. Initial program 41.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right)}}{2} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right)\right)}}{2} \]

      2. +-commutative N/A

         \[\leadsto \frac{x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + \frac{1}{60} \cdot {x}^{2}\right) + 2\right)}}{2} \]

      3. lower-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{3} + \frac{1}{60} \cdot {x}^{2}, 2\right)}}{2} \]

      4. unpow2 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} + \frac{1}{60} \cdot {x}^{2}, 2\right)}{2} \]

      5. lower-*.f64 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{3} + \frac{1}{60} \cdot {x}^{2}, 2\right)}{2} \]

      6. +-commutative N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\frac{1}{60} \cdot {x}^{2} + \frac{1}{3}}, 2\right)}{2} \]

      7. unpow2 N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, \frac{1}{60} \cdot \color{blue}{\left(x \cdot x\right)} + \frac{1}{3}, 2\right)}{2} \]

      8. associate-*r* N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\left(\frac{1}{60} \cdot x\right) \cdot x} + \frac{1}{3}, 2\right)}{2} \]

      9. *-commutative N/A

         \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{x \cdot \left(\frac{1}{60} \cdot x\right)} + \frac{1}{3}, 2\right)}{2} \]

      10. lower-fma.f64 N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, \color{blue}{\mathsf{fma}\left(x, \frac{1}{60} \cdot x, \frac{1}{3}\right)}, 2\right)}{2} \]

      11. *-commutative N/A

          \[\leadsto \frac{x \cdot \mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, \color{blue}{x \cdot \frac{1}{60}}, \frac{1}{3}\right), 2\right)}{2} \]

      12. lower-*.f64 86.5

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

   5. Simplified 86.5%

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

      1. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot \left(x \cdot \left(x \cdot \frac{1}{60}\right) + \frac{1}{3}\right) + 2\right)}{2} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot \color{blue}{\left(x \cdot \frac{1}{60}\right)} + \frac{1}{3}\right) + 2\right)}{2} \]

      3. lift-fma.f64 N/A

         \[\leadsto \frac{x \cdot \left(\left(x \cdot x\right) \cdot \color{blue}{\mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right)} + 2\right)}{2} \]

      4. lift-fma.f64 N/A

         \[\leadsto \frac{x \cdot \color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right), 2\right)}}{2} \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right), 2\right) \cdot x}}{2} \]

      6. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right) + 2\right)} \cdot x}{2} \]

      7. flip-+ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right)\right) - 2 \cdot 2}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right) - 2}} \cdot x}{2} \]

      8. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right)\right) - 2 \cdot 2\right) \cdot x}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right) - 2}}}{2} \]

      9. lower-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{\left(\left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right)\right) \cdot \left(\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right)\right) - 2 \cdot 2\right) \cdot x}{\left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \frac{1}{60}, \frac{1}{3}\right) - 2}}}{2} \]

   7. Applied egg-rr 73.9%

      \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right)\right), -4\right) \cdot x}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right), -2\right)}}}{2} \]

   if 4.99999999999999975e60 < x

   1. Initial program 100.0%

      \[\frac{e^{x} - e^{-x}}{2} \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. sinh-undef N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-sinh.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
   6. Step-by-step derivation

      1. distribute-rgt-in N/A

         \[\leadsto \color{blue}{1 \cdot x + \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

      2. *-lft-identity N/A

         \[\leadsto \color{blue}{x} + \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x + x} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}\right)} \cdot x + x \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot x\right)} + x \]

      6. unpow2 N/A

         \[\leadsto \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + x \]

      7. unpow3 N/A

         \[\leadsto \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{3}} + x \]

      8. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}, {x}^{3}, x\right)} \]

      9. +-commutative N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {x}^{2} + \frac{1}{6}}, {x}^{3}, x\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, {x}^{3}, x\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, {x}^{3}, x\right) \]

      12. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120}, \frac{1}{6}\right), {x}^{3}, x\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120}, \frac{1}{6}\right), {x}^{3}, x\right) \]

      14. cube-mult N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{1}{6}\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x\right) \]

      15. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{1}{6}\right), x \cdot \color{blue}{{x}^{2}}, x\right) \]

      16. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{1}{6}\right), \color{blue}{x \cdot {x}^{2}}, x\right) \]

      17. unpow2 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{1}{6}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

      18. lower-*.f64 100.0

          \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   7. Simplified 100.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 5 \cdot 10^{+60}:\\ \;\;\;\;\frac{\frac{x \cdot \mathsf{fma}\left(x \cdot x, \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right)\right) \cdot \left(x \cdot \mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right)\right), -4\right)}{\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x, x \cdot 0.016666666666666666, 0.3333333333333333\right), -2\right)}}{2}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.1% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma
  (fma
   x
   (* x (fma (* x x) 0.0001984126984126984 0.008333333333333333))
   0.16666666666666666)
  (* x (* x x))
  x))

C

double code(double x) {
	return fma(fma(x, (x * fma((x * x), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), (x * (x * x)), x);
}

Julia

function code(x)
	return fma(fma(x, Float64(x * fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333)), 0.16666666666666666), Float64(x * Float64(x * x)), x)
end

Wolfram

code[x_] := N[(N[(x * N[(x * N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]), $MachinePrecision] + 0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 52.7%

   \[\frac{e^{x} - e^{-x}}{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sinh-undef N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sinh.f64 100.0

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

4. Applied egg-rr 100.0%

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

   1. lift-sinh.f64 N/A

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

   2. associate-/l* N/A

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

   3. metadata-eval N/A

      \[\leadsto \sinh x \cdot \color{blue}{1} \]

   4. *-rgt-identity 100.0

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

6. Applied egg-rr 100.0%

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

7. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
8. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto x \cdot \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) + 1\right)} \]

   2. distribute-rgt-in N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right) \cdot x + 1 \cdot x} \]

   3. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot {x}^{2}\right)} \cdot x + 1 \cdot x \]

   4. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot \left({x}^{2} \cdot x\right)} + 1 \cdot x \]

   5. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + 1 \cdot x \]

   6. unpow3 N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot \color{blue}{{x}^{3}} + 1 \cdot x \]

   7. *-lft-identity N/A

      \[\leadsto \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right) \cdot {x}^{3} + \color{blue}{x} \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right), {x}^{3}, x\right)} \]

9. Simplified 92.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \]
10. Add Preprocessing

Alternative 5: 90.6% accurate, 7.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (fma (fma (* x x) 0.008333333333333333 0.16666666666666666) (* x (* x x)) x))

C

double code(double x) {
	return fma(fma((x * x), 0.008333333333333333, 0.16666666666666666), (x * (x * x)), x);
}

Julia

function code(x)
	return fma(fma(Float64(x * x), 0.008333333333333333, 0.16666666666666666), Float64(x * Float64(x * x)), x)
end

Wolfram

code[x_] := N[(N[(N[(x * x), $MachinePrecision] * 0.008333333333333333 + 0.16666666666666666), $MachinePrecision] * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 52.7%

   \[\frac{e^{x} - e^{-x}}{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sinh-undef N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sinh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. distribute-rgt-in N/A

      \[\leadsto \color{blue}{1 \cdot x + \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x} \]

   2. *-lft-identity N/A

      \[\leadsto \color{blue}{x} + \left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left({x}^{2} \cdot \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right)\right) \cdot x + x} \]

   4. *-commutative N/A

      \[\leadsto \color{blue}{\left(\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot {x}^{2}\right)} \cdot x + x \]

   5. associate-*l* N/A

      \[\leadsto \color{blue}{\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot \left({x}^{2} \cdot x\right)} + x \]

   6. unpow2 N/A

      \[\leadsto \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot \left(\color{blue}{\left(x \cdot x\right)} \cdot x\right) + x \]

   7. unpow3 N/A

      \[\leadsto \left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}\right) \cdot \color{blue}{{x}^{3}} + x \]

   8. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{1}{6} + \frac{1}{120} \cdot {x}^{2}, {x}^{3}, x\right)} \]

   9. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{1}{120} \cdot {x}^{2} + \frac{1}{6}}, {x}^{3}, x\right) \]

   10. *-commutative N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \frac{1}{120}} + \frac{1}{6}, {x}^{3}, x\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{fma}\left({x}^{2}, \frac{1}{120}, \frac{1}{6}\right)}, {x}^{3}, x\right) \]

   12. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120}, \frac{1}{6}\right), {x}^{3}, x\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{x \cdot x}, \frac{1}{120}, \frac{1}{6}\right), {x}^{3}, x\right) \]

   14. cube-mult N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{1}{6}\right), \color{blue}{x \cdot \left(x \cdot x\right)}, x\right) \]

   15. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{1}{6}\right), x \cdot \color{blue}{{x}^{2}}, x\right) \]

   16. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{1}{6}\right), \color{blue}{x \cdot {x}^{2}}, x\right) \]

   17. unpow2 N/A

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, \frac{1}{120}, \frac{1}{6}\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

   18. lower-*.f64 89.0

       \[\leadsto \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), x \cdot \color{blue}{\left(x \cdot x\right)}, x\right) \]

7. Simplified 89.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.008333333333333333, 0.16666666666666666\right), x \cdot \left(x \cdot x\right), x\right)} \]
8. Add Preprocessing

Alternative 6: 84.5% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, x \cdot \left(x \cdot 0.16666666666666666\right), x\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (fma x (* x (* x 0.16666666666666666)) x))

C

double code(double x) {
	return fma(x, (x * (x * 0.16666666666666666)), x);
}

Julia

function code(x)
	return fma(x, Float64(x * Float64(x * 0.16666666666666666)), x)
end

Wolfram

code[x_] := N[(x * N[(x * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]

Derivation

1. Initial program 52.7%

   \[\frac{e^{x} - e^{-x}}{2} \]
2. Add Preprocessing
3. Step-by-step derivation

   1. sinh-undef N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 N/A

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

   4. lower-sinh.f64 100.0

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

4. Applied egg-rr 100.0%

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

5. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \frac{1}{6} \cdot {x}^{2}, x\right)} \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{{x}^{2} \cdot \frac{1}{6}}, x\right) \]

   6. unpow2 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{\left(x \cdot x\right)} \cdot \frac{1}{6}, x\right) \]

   7. associate-*l* N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{1}{6}\right)}, x\right) \]

   8. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(x, \color{blue}{x \cdot \left(x \cdot \frac{1}{6}\right)}, x\right) \]

   9. lower-*.f64 82.6

      \[\leadsto \mathsf{fma}\left(x, x \cdot \color{blue}{\left(x \cdot 0.16666666666666666\right)}, x\right) \]

7. Simplified 82.6%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, x \cdot \left(x \cdot 0.16666666666666666\right), x\right)} \]
8. Add Preprocessing

Alternative 7: 52.7% accurate, 217.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x
end function

Java

public static double code(double x) {
	return x;
}

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

function tmp = code(x)
	tmp = x;
end

Wolfram

code[x_] := x

Derivation

1. Initial program 52.7%

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

3. Taylor expanded in x around 0

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

   1. lower-*.f64 53.8

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

5. Simplified 53.8%

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

   1. *-commutative N/A

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

   2. associate-/l* N/A

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

   3. metadata-eval N/A

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

   4. *-rgt-identity 53.8

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

7. Applied egg-rr 53.8%

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))