Hyperbolic sine

Percentage Accurate: 53.9% → 100.0%

Time: 7.5s

Alternatives: 7

Speedup: 9.9×

• 49.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 53.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 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 55.3%

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

   1. lift-/.f64 N/A

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

   2. lift--.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-exp.f64 N/A

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

   5. lift-neg.f64 N/A

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

   6. sinh-def-rev 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: 93.3% accurate, 4.9× speedup

Math

\[\begin{array}{l} \\ 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right) \cdot x, x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  0.5
  (*
   (fma
    (*
     (fma
      (* (fma 0.0003968253968253968 (* x x) 0.016666666666666666) x)
      x
      0.3333333333333333)
     x)
    x
    2.0)
   x)))

C

double code(double x) {
	return 0.5 * (fma((fma((fma(0.0003968253968253968, (x * x), 0.016666666666666666) * x), x, 0.3333333333333333) * x), x, 2.0) * x);
}

Julia

function code(x)
	return Float64(0.5 * Float64(fma(Float64(fma(Float64(fma(0.0003968253968253968, Float64(x * x), 0.016666666666666666) * x), x, 0.3333333333333333) * x), x, 2.0) * x))
end

Wolfram

code[x_] := N[(0.5 * N[(N[(N[(N[(N[(N[(0.0003968253968253968 * N[(x * x), $MachinePrecision] + 0.016666666666666666), $MachinePrecision] * x), $MachinePrecision] * x + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right)}}{2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right) \cdot x}}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right) \cdot x}}{2} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right) + 2\right)} \cdot x}{2} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 2\right) \cdot x}{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right), {x}^{2}, 2\right)} \cdot x}{2} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right) + \frac{1}{3}}, {x}^{2}, 2\right) \cdot x}{2} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{3}, {x}^{2}, 2\right) \cdot x}{2} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 2\right) \cdot x}{2} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{2520} \cdot {x}^{2} + \frac{1}{60}}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2520}, {x}^{2}, \frac{1}{60}\right)}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{x \cdot x}, \frac{1}{60}\right), {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{x \cdot x}, \frac{1}{60}\right), {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   15. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

   16. lower-*.f64 93.4

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

5. Applied rewrites 93.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x}{2}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. metadata-eval 93.4

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{0.5} \]

7. Applied rewrites 93.4%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 93.4%

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(x \cdot x, 0.0003968253968253968, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5 \]
10. Step-by-step derivation

11. Applied rewrites 93.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right) \cdot x, x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5 \]

12. Final simplification 93.4%

    \[\leadsto 0.5 \cdot \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right) \cdot x, x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \]
13. Add Preprocessing

Alternative 3: 93.1% accurate, 5.0× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.0003968253968253968, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (*
   (fma
    (fma (* (* x x) 0.0003968253968253968) (* x x) 0.3333333333333333)
    (* x x)
    2.0)
   x)
  0.5))

C

double code(double x) {
	return (fma(fma(((x * x) * 0.0003968253968253968), (x * x), 0.3333333333333333), (x * x), 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(fma(Float64(Float64(x * x) * 0.0003968253968253968), Float64(x * x), 0.3333333333333333), Float64(x * x), 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(N[(x * x), $MachinePrecision] * 0.0003968253968253968), $MachinePrecision] * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * N[(x * x), $MachinePrecision] + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 55.3%

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

3. Taylor expanded in x around 0

   \[\leadsto \frac{\color{blue}{x \cdot \left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right)}}{2} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right) \cdot x}}{2} \]

   2. lower-*.f64 N/A

      \[\leadsto \frac{\color{blue}{\left(2 + {x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right)\right) \cdot x}}{2} \]

   3. +-commutative N/A

      \[\leadsto \frac{\color{blue}{\left({x}^{2} \cdot \left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right) + 2\right)} \cdot x}{2} \]

   4. *-commutative N/A

      \[\leadsto \frac{\left(\color{blue}{\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right)\right) \cdot {x}^{2}} + 2\right) \cdot x}{2} \]

   5. lower-fma.f64 N/A

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{1}{3} + {x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right), {x}^{2}, 2\right)} \cdot x}{2} \]

   6. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{x}^{2} \cdot \left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right) + \frac{1}{3}}, {x}^{2}, 2\right) \cdot x}{2} \]

   7. *-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}\right) \cdot {x}^{2}} + \frac{1}{3}, {x}^{2}, 2\right) \cdot x}{2} \]

   8. lower-fma.f64 N/A

      \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{60} + \frac{1}{2520} \cdot {x}^{2}, {x}^{2}, \frac{1}{3}\right)}, {x}^{2}, 2\right) \cdot x}{2} \]

   9. +-commutative N/A

      \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\frac{1}{2520} \cdot {x}^{2} + \frac{1}{60}}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   10. lower-fma.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\color{blue}{\mathsf{fma}\left(\frac{1}{2520}, {x}^{2}, \frac{1}{60}\right)}, {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   11. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{x \cdot x}, \frac{1}{60}\right), {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   12. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, \color{blue}{x \cdot x}, \frac{1}{60}\right), {x}^{2}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   13. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   14. lower-*.f64 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), \color{blue}{x \cdot x}, \frac{1}{3}\right), {x}^{2}, 2\right) \cdot x}{2} \]

   15. unpow2 N/A

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

   16. lower-*.f64 93.4

       \[\leadsto \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), \color{blue}{x \cdot x}, 2\right) \cdot x}{2} \]

5. Applied rewrites 93.4%

   \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x}}{2} \]
6. Step-by-step derivation

   1. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x}{2}} \]

   2. div-inv N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   3. lower-*.f64 N/A

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520}, x \cdot x, \frac{1}{60}\right), x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2}} \]

   4. metadata-eval 93.4

      \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot \color{blue}{0.5} \]

7. Applied rewrites 93.4%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968, x \cdot x, 0.016666666666666666\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5} \]

8. Taylor expanded in x around inf

   \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\frac{1}{2520} \cdot {x}^{2}, x \cdot x, \frac{1}{3}\right), x \cdot x, 2\right) \cdot x\right) \cdot \frac{1}{2} \]
9. Step-by-step derivation

10. Applied rewrites 93.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(0.0003968253968253968 \cdot \left(x \cdot x\right), x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \]

11. Final simplification 93.4%

    \[\leadsto \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(x \cdot x\right) \cdot 0.0003968253968253968, x \cdot x, 0.3333333333333333\right), x \cdot x, 2\right) \cdot x\right) \cdot 0.5 \]
12. Add Preprocessing

Alternative 4: 90.8% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  (* (fma (* (fma 0.016666666666666666 (* x x) 0.3333333333333333) x) x 2.0) x)
  0.5))

C

double code(double x) {
	return (fma((fma(0.016666666666666666, (x * x), 0.3333333333333333) * x), x, 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(fma(0.016666666666666666, Float64(x * x), 0.3333333333333333) * x), x, 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.016666666666666666 * N[(x * x), $MachinePrecision] + 0.3333333333333333), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 92.2

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

5. Applied rewrites 92.2%

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

7. Applied rewrites 92.2%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 92.2

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5} \]

9. Applied rewrites 92.2%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5} \]
10. Add Preprocessing

Alternative 5: 90.4% accurate, 6.8× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(\left(0.016666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (fma (* (* 0.016666666666666666 (* x x)) x) x 2.0) x) 0.5))

C

double code(double x) {
	return (fma(((0.016666666666666666 * (x * x)) * x), x, 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(Float64(0.016666666666666666 * Float64(x * x)) * x), x, 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(N[(0.016666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. +-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. unpow2 N/A

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

   9. lower-*.f64 N/A

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

   10. unpow2 N/A

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

   11. lower-*.f64 92.2

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

5. Applied rewrites 92.2%

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

7. Applied rewrites 92.2%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 92.2

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5} \]

9. Applied rewrites 92.2%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(0.016666666666666666, x \cdot x, 0.3333333333333333\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5} \]

10. Taylor expanded in x around inf

    \[\leadsto \left(\mathsf{fma}\left(\left(\frac{1}{60} \cdot {x}^{2}\right) \cdot x, x, 2\right) \cdot x\right) \cdot \frac{1}{2} \]
11. Step-by-step derivation

12. Applied rewrites 92.2%

    \[\leadsto \left(\mathsf{fma}\left(\left(0.016666666666666666 \cdot \left(x \cdot x\right)\right) \cdot x, x, 2\right) \cdot x\right) \cdot 0.5 \]
13. Add Preprocessing

Alternative 6: 84.4% accurate, 9.9× speedup

Math

\[\begin{array}{l} \\ \left(\mathsf{fma}\left(0.3333333333333333 \cdot x, x, 2\right) \cdot x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* (* (fma (* 0.3333333333333333 x) x 2.0) x) 0.5))

C

double code(double x) {
	return (fma((0.3333333333333333 * x), x, 2.0) * x) * 0.5;
}

Julia

function code(x)
	return Float64(Float64(fma(Float64(0.3333333333333333 * x), x, 2.0) * x) * 0.5)
end

Wolfram

code[x_] := N[(N[(N[(N[(0.3333333333333333 * x), $MachinePrecision] * x + 2.0), $MachinePrecision] * x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{e^{x} - e^{-x}}{2} \]
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. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. +-commutative N/A

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

   4. lower-fma.f64 N/A

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

   5. unpow2 N/A

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

   6. lower-*.f64 86.7

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

5. Applied rewrites 86.7%

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

7. Applied rewrites 86.7%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. metadata-eval N/A

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

   4. lower-*.f64 86.7

      \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333 \cdot x, x, 2\right) \cdot x\right) \cdot 0.5} \]

9. Applied rewrites 86.7%

   \[\leadsto \color{blue}{\left(\mathsf{fma}\left(0.3333333333333333 \cdot x, x, 2\right) \cdot x\right) \cdot 0.5} \]
10. Add Preprocessing

Alternative 7: 52.6% accurate, 24.1× speedup

Math

\[\begin{array}{l} \\ \left(x + x\right) \cdot 0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (+ x x) 0.5))

C

double code(double x) {
	return (x + x) * 0.5;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x + x) * 0.5d0
end function

Java

public static double code(double x) {
	return (x + x) * 0.5;
}

Python

def code(x):
	return (x + x) * 0.5

Julia

function code(x)
	return Float64(Float64(x + x) * 0.5)
end

MATLAB

function tmp = code(x)
	tmp = (x + x) * 0.5;
end

Wolfram

code[x_] := N[(N[(x + x), $MachinePrecision] * 0.5), $MachinePrecision]

Derivation

1. Initial program 55.3%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 51.3

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

5. Applied rewrites 51.3%

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

   1. lift-/.f64 N/A

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

   2. div-inv N/A

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

   3. lower-*.f64 N/A

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

   4. metadata-eval 51.3

      \[\leadsto \left(2 \cdot x\right) \cdot \color{blue}{0.5} \]

7. Applied rewrites 51.3%

   \[\leadsto \color{blue}{\left(2 \cdot x\right) \cdot 0.5} \]
8. Step-by-step derivation

9. Applied rewrites 51.3%

   \[\leadsto \left(x + \color{blue}{x}\right) \cdot 0.5 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024298 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))