Hyperbolic sine

Percentage Accurate: 54.3% → 100.0%

Time: 4.4s

Alternatives: 2

Speedup: 2.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 54.3% 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, 2.0× 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 54.9%

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

   1. div-sub 54.9%

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

   2. metadata-eval 54.9%

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{2}{1}}} - \frac{e^{-x}}{2} \]

   3. associate-/l* 54.9%

      \[\leadsto \color{blue}{\frac{e^{x} \cdot 1}{2}} - \frac{e^{-x}}{2} \]

   4. *-commutative 54.9%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{x}}}{2} - \frac{e^{-x}}{2} \]

   5. associate-/l* 54.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{e^{x}}}} - \frac{e^{-x}}{2} \]

   6. associate-/r/ 54.9%

      \[\leadsto \color{blue}{\frac{1}{2} \cdot e^{x}} - \frac{e^{-x}}{2} \]

   7. *-commutative 54.9%

      \[\leadsto \color{blue}{e^{x} \cdot \frac{1}{2}} - \frac{e^{-x}}{2} \]

   8. fma-neg 54.9%

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

   9. metadata-eval 54.9%

      \[\leadsto \mathsf{fma}\left(e^{x}, \color{blue}{0.5}, -\frac{e^{-x}}{2}\right) \]

   10. exp-neg 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, -\frac{\color{blue}{\frac{1}{e^{x}}}}{2}\right) \]

   11. associate-/l/ 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, -\color{blue}{\frac{1}{2 \cdot e^{x}}}\right) \]

   12. distribute-neg-frac 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, \color{blue}{\frac{-1}{2 \cdot e^{x}}}\right) \]

   13. metadata-eval 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, \frac{\color{blue}{-1}}{2 \cdot e^{x}}\right) \]

   14. associate-/r* 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, \color{blue}{\frac{\frac{-1}{2}}{e^{x}}}\right) \]

   15. metadata-eval 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, \frac{\color{blue}{-0.5}}{e^{x}}\right) \]

3. Simplified 54.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{x}, 0.5, \frac{-0.5}{e^{x}}\right)} \]

4. Taylor expanded in x around inf 54.8%

   \[\leadsto \color{blue}{0.5 \cdot e^{x} - 0.5 \cdot \frac{1}{e^{x}}} \]
5. Step-by-step derivation

   1. distribute-lft-out-- 54.8%

      \[\leadsto \color{blue}{0.5 \cdot \left(e^{x} - \frac{1}{e^{x}}\right)} \]

   2. rec-exp 54.9%

      \[\leadsto 0.5 \cdot \left(e^{x} - \color{blue}{e^{-x}}\right) \]

   3. sinh-undef 100.0%

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

6. Applied egg-rr 100.0%

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

   1. associate-*r* 100.0%

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

   2. metadata-eval 100.0%

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

   3. *-lft-identity 100.0%

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

8. Simplified 100.0%

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

9. Final simplification 100.0%

   \[\leadsto \sinh x \]

Alternative 2: 52.2% accurate, 206.0× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 x)

C

double code(double x) {
	return x;
}

Fortran

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

Java

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

Python

def code(x):
	return x

Julia

function code(x)
	return x
end

MATLAB

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

Wolfram

code[x_] := x

Derivation

1. Initial program 54.9%

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

   1. div-sub 54.9%

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

   2. metadata-eval 54.9%

      \[\leadsto \frac{e^{x}}{\color{blue}{\frac{2}{1}}} - \frac{e^{-x}}{2} \]

   3. associate-/l* 54.9%

      \[\leadsto \color{blue}{\frac{e^{x} \cdot 1}{2}} - \frac{e^{-x}}{2} \]

   4. *-commutative 54.9%

      \[\leadsto \frac{\color{blue}{1 \cdot e^{x}}}{2} - \frac{e^{-x}}{2} \]

   5. associate-/l* 54.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{2}{e^{x}}}} - \frac{e^{-x}}{2} \]

   6. associate-/r/ 54.9%

      \[\leadsto \color{blue}{\frac{1}{2} \cdot e^{x}} - \frac{e^{-x}}{2} \]

   7. *-commutative 54.9%

      \[\leadsto \color{blue}{e^{x} \cdot \frac{1}{2}} - \frac{e^{-x}}{2} \]

   8. fma-neg 54.9%

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

   9. metadata-eval 54.9%

      \[\leadsto \mathsf{fma}\left(e^{x}, \color{blue}{0.5}, -\frac{e^{-x}}{2}\right) \]

   10. exp-neg 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, -\frac{\color{blue}{\frac{1}{e^{x}}}}{2}\right) \]

   11. associate-/l/ 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, -\color{blue}{\frac{1}{2 \cdot e^{x}}}\right) \]

   12. distribute-neg-frac 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, \color{blue}{\frac{-1}{2 \cdot e^{x}}}\right) \]

   13. metadata-eval 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, \frac{\color{blue}{-1}}{2 \cdot e^{x}}\right) \]

   14. associate-/r* 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, \color{blue}{\frac{\frac{-1}{2}}{e^{x}}}\right) \]

   15. metadata-eval 54.8%

       \[\leadsto \mathsf{fma}\left(e^{x}, 0.5, \frac{\color{blue}{-0.5}}{e^{x}}\right) \]

3. Simplified 54.8%

   \[\leadsto \color{blue}{\mathsf{fma}\left(e^{x}, 0.5, \frac{-0.5}{e^{x}}\right)} \]

4. Taylor expanded in x around 0 51.4%

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

5. Final simplification 51.4%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023320 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))