Hyperbolic sine

Percentage Accurate: 54.2% → 100.0%

Time: 10.0s

Alternatives: 10

Speedup: 12.8×

• 49.5% 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: 54.2% 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 54.6%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. sinh-def N/A

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

   7. lower-sinh.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 87.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{x} - e^{-x} \leq 10:\\ \;\;\;\;\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot 0.16666666666666666, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (- (exp x) (exp (- x))) 10.0)
   (fma x (* (* x x) 0.16666666666666666) x)
   (* x (* 0.008333333333333333 (* x (* x (* x x)))))))

C

double code(double x) {
	double tmp;
	if ((exp(x) - exp(-x)) <= 10.0) {
		tmp = fma(x, ((x * x) * 0.16666666666666666), x);
	} else {
		tmp = x * (0.008333333333333333 * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	tmp = 0.0
	if (Float64(exp(x) - exp(Float64(-x))) <= 10.0)
		tmp = fma(x, Float64(Float64(x * x) * 0.16666666666666666), x);
	else
		tmp = Float64(x * Float64(0.008333333333333333 * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := If[LessEqual[N[(N[Exp[x], $MachinePrecision] - N[Exp[(-x)], $MachinePrecision]), $MachinePrecision], 10.0], N[(x * N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(0.008333333333333333 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (exp.f64 x) (exp.f64 (neg.f64 x))) < 10

   1. Initial program 40.7%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right)} \]
   4. 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. lower-*.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 87.3

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

   5. Applied rewrites 87.3%

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

   if 10 < (-.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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{120} \cdot \color{blue}{{x}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 79.4%

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

4. Final simplification 85.4%

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

Alternative 3: 76.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x \cdot \left(x \cdot x\right)\\ t_1 := \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\\ t_2 := x \cdot \left(x \cdot t\_1\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{+61}:\\ \;\;\;\;x + \frac{t\_0 \cdot \mathsf{fma}\left(t\_2, t\_2, -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, t\_1, -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.008333333333333333 \cdot \left(x \cdot t\_0\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (* x (* x x)))
        (t_1 (fma (* x x) 0.0001984126984126984 0.008333333333333333))
        (t_2 (* x (* x t_1))))
   (if (<= x 4e+61)
     (+
      x
      (/
       (* t_0 (fma t_2 t_2 -0.027777777777777776))
       (fma (* x x) t_1 -0.16666666666666666)))
     (* x (* 0.008333333333333333 (* x t_0))))))

C

double code(double x) {
	double t_0 = x * (x * x);
	double t_1 = fma((x * x), 0.0001984126984126984, 0.008333333333333333);
	double t_2 = x * (x * t_1);
	double tmp;
	if (x <= 4e+61) {
		tmp = x + ((t_0 * fma(t_2, t_2, -0.027777777777777776)) / fma((x * x), t_1, -0.16666666666666666));
	} else {
		tmp = x * (0.008333333333333333 * (x * t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(x * Float64(x * x))
	t_1 = fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333)
	t_2 = Float64(x * Float64(x * t_1))
	tmp = 0.0
	if (x <= 4e+61)
		tmp = Float64(x + Float64(Float64(t_0 * fma(t_2, t_2, -0.027777777777777776)) / fma(Float64(x * x), t_1, -0.16666666666666666)));
	else
		tmp = Float64(x * Float64(0.008333333333333333 * Float64(x * t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(x * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 4e+61], N[(x + N[(N[(t$95$0 * N[(t$95$2 * t$95$2 + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * t$95$1 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(0.008333333333333333 * N[(x * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999998e61

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. 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-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 91.0%

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

   9. Applied rewrites 72.5%

      \[\leadsto \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\right), x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)} + x \]

   if 3.9999999999999998e61 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{120} \cdot \color{blue}{{x}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+61}:\\ \;\;\;\;x + \frac{\left(x \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\right), x \cdot \left(x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.7% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right)\\ \mathbf{if}\;x \leq 4 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(t\_0 \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), t\_0, -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, t\_0, -0.16666666666666666\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma (* x x) 0.0001984126984126984 0.008333333333333333)))
   (if (<= x 4e+61)
     (fma
      x
      (/
       (* (* x x) (fma (* t_0 (* (* x x) (* x x))) t_0 -0.027777777777777776))
       (fma (* x x) t_0 -0.16666666666666666))
      x)
     (* x (* 0.008333333333333333 (* x (* x (* x x))))))))

C

double code(double x) {
	double t_0 = fma((x * x), 0.0001984126984126984, 0.008333333333333333);
	double tmp;
	if (x <= 4e+61) {
		tmp = fma(x, (((x * x) * fma((t_0 * ((x * x) * (x * x))), t_0, -0.027777777777777776)) / fma((x * x), t_0, -0.16666666666666666)), x);
	} else {
		tmp = x * (0.008333333333333333 * (x * (x * (x * x))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(Float64(x * x), 0.0001984126984126984, 0.008333333333333333)
	tmp = 0.0
	if (x <= 4e+61)
		tmp = fma(x, Float64(Float64(Float64(x * x) * fma(Float64(t_0 * Float64(Float64(x * x) * Float64(x * x))), t_0, -0.027777777777777776)) / fma(Float64(x * x), t_0, -0.16666666666666666)), x);
	else
		tmp = Float64(x * Float64(0.008333333333333333 * Float64(x * Float64(x * Float64(x * x)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x * x), $MachinePrecision] * 0.0001984126984126984 + 0.008333333333333333), $MachinePrecision]}, If[LessEqual[x, 4e+61], N[(x * N[(N[(N[(x * x), $MachinePrecision] * N[(N[(t$95$0 * N[(N[(x * x), $MachinePrecision] * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * t$95$0 + -0.027777777777777776), $MachinePrecision]), $MachinePrecision] / N[(N[(x * x), $MachinePrecision] * t$95$0 + -0.16666666666666666), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x * N[(0.008333333333333333 * N[(x * N[(x * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 3.9999999999999998e61

   1. Initial program 44.4%

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

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
   4. 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-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

   5. Applied rewrites 91.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x\right)} \]
   6. Step-by-step derivation

   7. Applied rewrites 72.1%

      \[\leadsto \mathsf{fma}\left(x, \frac{\mathsf{fma}\left(\left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right) \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), -0.027777777777777776\right) \cdot \left(x \cdot x\right)}{\color{blue}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)}}, x\right) \]

   if 3.9999999999999998e61 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-commutative N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. *-rgt-identity N/A

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

      7. lower-fma.f64 N/A

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

      8. unpow2 N/A

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

      9. lower-*.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. +-commutative N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

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

      14. unpow2 N/A

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

      15. lower-*.f64 100.0

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

   5. Applied rewrites 100.0%

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

   6. Taylor expanded in x around inf

      \[\leadsto \frac{1}{120} \cdot \color{blue}{{x}^{5}} \]
   7. Step-by-step derivation

   8. Applied rewrites 100.0%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+61}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{\left(x \cdot x\right) \cdot \mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right) \cdot \left(\left(x \cdot x\right) \cdot \left(x \cdot x\right)\right), \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), -0.027777777777777776\right)}{\mathsf{fma}\left(x \cdot x, \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), -0.16666666666666666\right)}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(0.008333333333333333 \cdot \left(x \cdot \left(x \cdot \left(x \cdot x\right)\right)\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.2% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. 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-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 92.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 92.7%

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

Alternative 6: 93.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(\frac{1}{6} + {x}^{2} \cdot \left(\frac{1}{120} + \frac{1}{5040} \cdot {x}^{2}\right)\right)\right)} \]
4. 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-lft-in N/A

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

   3. *-rgt-identity N/A

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

   4. lower-fma.f64 N/A

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

5. Applied rewrites 92.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, x \cdot \mathsf{fma}\left(x \cdot x, 0.0001984126984126984, 0.008333333333333333\right), 0.16666666666666666\right), x\right)} \]
6. Step-by-step derivation

7. Applied rewrites 92.7%

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

8. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, \frac{1}{5040} \cdot \color{blue}{{x}^{3}}, \frac{1}{6}\right), x\right) \]
9. Step-by-step derivation

10. Applied rewrites 92.1%

    \[\leadsto \mathsf{fma}\left(x, \left(x \cdot x\right) \cdot \mathsf{fma}\left(x, 0.0001984126984126984 \cdot \color{blue}{\left(x \cdot \left(x \cdot x\right)\right)}, 0.16666666666666666\right), x\right) \]

11. Final simplification 92.1%

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

Alternative 7: 90.4% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 89.5

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

5. Applied rewrites 89.5%

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

Alternative 8: 90.0% accurate, 8.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. *-commutative N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. *-rgt-identity N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. +-commutative N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

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

   14. unpow2 N/A

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

   15. lower-*.f64 89.5

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

5. Applied rewrites 89.5%

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

6. Taylor expanded in x around inf

   \[\leadsto \mathsf{fma}\left(x \cdot x, \frac{1}{120} \cdot \color{blue}{{x}^{3}}, x\right) \]
7. Step-by-step derivation

8. Applied rewrites 88.5%

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

9. Final simplification 88.5%

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

Alternative 9: 84.1% accurate, 12.8× speedup

Math

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

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(Float64(x * x) * 0.16666666666666666), x)
end

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right)} \]
4. 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. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 81.0

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

5. Applied rewrites 81.0%

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

6. Final simplification 81.0%

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

Alternative 10: 37.5% accurate, 13.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x * (x * x)) * 0.16666666666666666d0
end function

Java

public static double code(double x) {
	return (x * (x * x)) * 0.16666666666666666;
}

Python

def code(x):
	return (x * (x * x)) * 0.16666666666666666

Julia

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

MATLAB

function tmp = code(x)
	tmp = (x * (x * x)) * 0.16666666666666666;
end

Wolfram

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

Derivation

1. Initial program 54.6%

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

3. Taylor expanded in x around 0

   \[\leadsto \color{blue}{x \cdot \left(1 + \frac{1}{6} \cdot {x}^{2}\right)} \]
4. 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. lower-*.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 81.0

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

5. Applied rewrites 81.0%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 34.4%

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

9. Final simplification 34.4%

   \[\leadsto \left(x \cdot \left(x \cdot x\right)\right) \cdot 0.16666666666666666 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))