Hyperbolic sine

Percentage Accurate: 54.1% → 100.0%

Time: 6.8s

Alternatives: 8

Speedup: 7.8×

• 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: 54.1% 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, 1.9× speedup

Math

\[\begin{array}{l} \\ \frac{2 \cdot \sinh x}{2} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (* 2.0 (sinh x)) 2.0))

C

double code(double x) {
	return (2.0 * sinh(x)) / 2.0;
}

Fortran

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

Java

public static double code(double x) {
	return (2.0 * Math.sinh(x)) / 2.0;
}

Python

def code(x):
	return (2.0 * math.sinh(x)) / 2.0

Julia

function code(x)
	return Float64(Float64(2.0 * sinh(x)) / 2.0)
end

MATLAB

function tmp = code(x)
	tmp = (2.0 * sinh(x)) / 2.0;
end

Wolfram

code[x_] := N[(N[(2.0 * N[Sinh[x], $MachinePrecision]), $MachinePrecision] / 2.0), $MachinePrecision]

Derivation

1. Initial program 54.3%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-exp.f64 N/A

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

   4. lift-neg.f64 N/A

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

   5. sinh-undef N/A

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

   6. *-commutative N/A

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

   7. lower-*.f64 N/A

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

   8. lower-sinh.f64 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

   \[\leadsto \frac{2 \cdot \sinh x}{2} \]
6. Add Preprocessing

Alternative 2: 93.0% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.3%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 93.5

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

5. Applied rewrites 93.5%

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

7. Applied rewrites 93.5%

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

9. Applied rewrites 93.5%

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

11. Applied rewrites 93.5%

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

Alternative 3: 92.9% accurate, 4.4× speedup

Math

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

FPCore

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

C

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

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) / 2.0)
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] / 2.0), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 93.5

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

5. Applied rewrites 93.5%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 93.2%

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

9. Final simplification 93.2%

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

Alternative 4: 90.4% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

function code(x)
	return Float64(Float64(fma(Float64(fma(0.016666666666666666, Float64(x * x), 0.3333333333333333) * x), x, 2.0) * x) / 2.0)
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] / 2.0), $MachinePrecision]

Derivation

1. Initial program 54.3%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 91.6

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

5. Applied rewrites 91.6%

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

7. Applied rewrites 91.6%

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

Alternative 5: 90.0% accurate, 5.7× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.3%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 91.6

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

5. Applied rewrites 91.6%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 91.2%

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

Alternative 6: 68.3% accurate, 6.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;0.5 \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x}{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.5) (* 0.5 (+ x x)) (/ (* (* 0.3333333333333333 (* x x)) x) 2.0)))

C

double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = 0.5 * (x + x);
	} else {
		tmp = ((0.3333333333333333 * (x * x)) * x) / 2.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = 0.5d0 * (x + x)
    else
        tmp = ((0.3333333333333333d0 * (x * x)) * x) / 2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = 0.5 * (x + x);
	} else {
		tmp = ((0.3333333333333333 * (x * x)) * x) / 2.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.5:
		tmp = 0.5 * (x + x)
	else:
		tmp = ((0.3333333333333333 * (x * x)) * x) / 2.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(0.5 * Float64(x + x));
	else
		tmp = Float64(Float64(Float64(0.3333333333333333 * Float64(x * x)) * x) / 2.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = 0.5 * (x + x);
	else
		tmp = ((0.3333333333333333 * (x * x)) * x) / 2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.5], N[(0.5 * N[(x + x), $MachinePrecision]), $MachinePrecision], N[(N[(N[(0.3333333333333333 * N[(x * x), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / 2.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 38.8%

      \[\frac{e^{x} - e^{-x}}{2} \]
   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 67.3

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

   5. Applied rewrites 67.3%

      \[\leadsto \frac{\color{blue}{2 \cdot x}}{2} \]
   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 67.3

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

   7. Applied rewrites 67.3%

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

   9. Applied rewrites 67.3%

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

   if 2.5 < 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. *-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. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. unpow2 N/A

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

      7. lower-*.f64 73.8

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

   5. Applied rewrites 73.8%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 73.8%

      \[\leadsto \frac{\left(0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x}{2} \]
3. Recombined 2 regimes into one program.

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;0.5 \cdot \left(x + x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(0.3333333333333333 \cdot \left(x \cdot x\right)\right) \cdot x}{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.1% accurate, 7.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.3%

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

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

   5. lower-fma.f64 N/A

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

   6. unpow2 N/A

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

   7. lower-*.f64 83.6

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

5. Applied rewrites 83.6%

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

7. Applied rewrites 83.6%

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

Alternative 8: 52.4% accurate, 24.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.3%

   \[\frac{e^{x} - e^{-x}}{2} \]
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 51.7

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

5. Applied rewrites 51.7%

   \[\leadsto \frac{\color{blue}{2 \cdot x}}{2} \]
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 51.7

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

7. Applied rewrites 51.7%

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

9. Applied rewrites 51.7%

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

10. Final simplification 51.7%

    \[\leadsto 0.5 \cdot \left(x + x\right) \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024312 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))