Hyperbolic sine

Percentage Accurate: 54.5% → 100.0%

Time: 9.3s

Alternatives: 6

Speedup: 29.4×

• 49.8% 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: 54.5% 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 57.8%

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

   1. div-sub 57.8%

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

   2. *-rgt-identity 57.8%

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

   3. associate-*l/ 57.8%

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

   4. associate-/l* 57.8%

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

   5. fma-neg 57.8%

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

   6. metadata-eval 57.8%

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

   7. exp-neg 57.8%

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

   8. associate-/l/ 57.8%

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

   9. associate-/r* 57.8%

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

   10. distribute-neg-frac 57.8%

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

   11. metadata-eval 57.8%

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

   12. metadata-eval 57.8%

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

3. Simplified 57.8%

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

5. Taylor expanded in x around inf 57.8%

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

   1. *-un-lft-identity 57.8%

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

   2. distribute-lft-out-- 57.8%

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

   3. rec-exp 57.8%

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

   4. sinh-undef 100.0%

      \[\leadsto 1 \cdot \left(0.5 \cdot \color{blue}{\left(2 \cdot \sinh x\right)}\right) \]

7. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{1 \cdot \left(0.5 \cdot \left(2 \cdot \sinh x\right)\right)} \]
8. Step-by-step derivation

   1. *-lft-identity 100.0%

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

   2. associate-*r* 100.0%

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

   3. metadata-eval 100.0%

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

   4. *-lft-identity 100.0%

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

9. Simplified 100.0%

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

Alternative 2: 90.1% accurate, 13.7× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (*
  x
  (+
   1.0
   (* (* x x) (+ 0.16666666666666666 (* (* x x) 0.008333333333333333))))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * (0.16666666666666666d0 + ((x * x) * 0.008333333333333333d0))))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333))));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * Float64(0.16666666666666666 + Float64(Float64(x * x) * 0.008333333333333333)))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * (0.16666666666666666 + ((x * x) * 0.008333333333333333))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * N[(0.16666666666666666 + N[(N[(x * x), $MachinePrecision] * 0.008333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

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

   1. div-sub 57.8%

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

   2. *-rgt-identity 57.8%

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

   3. associate-*l/ 57.8%

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

   4. associate-/l* 57.8%

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

   5. fma-neg 57.8%

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

   6. metadata-eval 57.8%

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

   7. exp-neg 57.8%

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

   8. associate-/l/ 57.8%

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

   9. associate-/r* 57.8%

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

   10. distribute-neg-frac 57.8%

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

   11. metadata-eval 57.8%

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

   12. metadata-eval 57.8%

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

3. Simplified 57.8%

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

5. Taylor expanded in x around 0 89.5%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{2}\right)\right)} \]
6. Step-by-step derivation

   1. distribute-lft-in 89.5%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left({x}^{2} \cdot 0.16666666666666666 + {x}^{2} \cdot \left(0.008333333333333333 \cdot {x}^{2}\right)\right)}\right) \]

   2. *-commutative 89.5%

      \[\leadsto x \cdot \left(1 + \left(\color{blue}{0.16666666666666666 \cdot {x}^{2}} + {x}^{2} \cdot \left(0.008333333333333333 \cdot {x}^{2}\right)\right)\right) \]

   3. associate-+r+ 89.5%

      \[\leadsto x \cdot \color{blue}{\left(\left(1 + 0.16666666666666666 \cdot {x}^{2}\right) + {x}^{2} \cdot \left(0.008333333333333333 \cdot {x}^{2}\right)\right)} \]

   4. associate-+r+ 89.5%

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(0.16666666666666666 \cdot {x}^{2} + {x}^{2} \cdot \left(0.008333333333333333 \cdot {x}^{2}\right)\right)\right)} \]

   5. *-commutative 89.5%

      \[\leadsto x \cdot \left(1 + \left(\color{blue}{{x}^{2} \cdot 0.16666666666666666} + {x}^{2} \cdot \left(0.008333333333333333 \cdot {x}^{2}\right)\right)\right) \]

   6. distribute-lft-in 89.5%

      \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot \left(0.16666666666666666 + 0.008333333333333333 \cdot {x}^{2}\right)}\right) \]

   7. *-commutative 89.5%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{{x}^{2} \cdot 0.008333333333333333}\right)\right) \]

7. Simplified 89.5%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + {x}^{2} \cdot 0.008333333333333333\right)\right)} \]
8. Step-by-step derivation

   1. unpow2 89.5%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333\right)\right) \]

9. Applied egg-rr 89.5%

   \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333\right)\right) \]
10. Step-by-step derivation

    1. unpow2 89.5%

       \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333\right)\right) \]

11. Applied egg-rr 89.5%

    \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot \left(0.16666666666666666 + \left(x \cdot x\right) \cdot 0.008333333333333333\right)\right) \]
12. Add Preprocessing

Alternative 3: 67.4% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(x \cdot 0.020833333333333332\right)\right)\right) \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (* x (+ 1.0 (* x (+ 0.25 (* x (* x 0.020833333333333332)))))))

C

double code(double x) {
	return x * (1.0 + (x * (0.25 + (x * (x * 0.020833333333333332)))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * (0.25d0 + (x * (x * 0.020833333333333332d0)))))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * (0.25 + (x * (x * 0.020833333333333332)))));
}

Python

def code(x):
	return x * (1.0 + (x * (0.25 + (x * (x * 0.020833333333333332)))))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * Float64(0.25 + Float64(x * Float64(x * 0.020833333333333332))))))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * (0.25 + (x * (x * 0.020833333333333332)))));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * N[(0.25 + N[(x * N[(x * 0.020833333333333332), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in x around 0 29.0%

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

   1. mul-1-neg 29.0%

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

   2. unsub-neg 29.0%

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

5. Simplified 29.0%

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

6. Taylor expanded in x around 0 66.3%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(0.08333333333333333 + 0.020833333333333332 \cdot x\right)\right)\right)} \]
7. Step-by-step derivation

   1. *-commutative 66.3%

      \[\leadsto x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(0.08333333333333333 + \color{blue}{x \cdot 0.020833333333333332}\right)\right)\right) \]

8. Simplified 66.3%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(0.08333333333333333 + x \cdot 0.020833333333333332\right)\right)\right)} \]

9. Taylor expanded in x around inf 66.3%

   \[\leadsto x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \color{blue}{\left(x \cdot \left(0.020833333333333332 + 0.08333333333333333 \cdot \frac{1}{x}\right)\right)}\right)\right) \]

10. Taylor expanded in x around inf 66.3%

    \[\leadsto x \cdot \left(1 + x \cdot \left(0.25 + x \cdot \left(x \cdot \color{blue}{0.020833333333333332}\right)\right)\right) \]
11. Add Preprocessing

Alternative 4: 83.6% accurate, 22.9× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + \left(x \cdot x\right) \cdot 0.16666666666666666\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 1.0 (* (* x x) 0.16666666666666666))))

C

double code(double x) {
	return x * (1.0 + ((x * x) * 0.16666666666666666));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + ((x * x) * 0.16666666666666666d0))
end function

Java

public static double code(double x) {
	return x * (1.0 + ((x * x) * 0.16666666666666666));
}

Python

def code(x):
	return x * (1.0 + ((x * x) * 0.16666666666666666))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(Float64(x * x) * 0.16666666666666666)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + ((x * x) * 0.16666666666666666));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(N[(x * x), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

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

   1. div-sub 57.8%

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

   2. *-rgt-identity 57.8%

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

   3. associate-*l/ 57.8%

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

   4. associate-/l* 57.8%

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

   5. fma-neg 57.8%

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

   6. metadata-eval 57.8%

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

   7. exp-neg 57.8%

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

   8. associate-/l/ 57.8%

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

   9. associate-/r* 57.8%

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

   10. distribute-neg-frac 57.8%

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

   11. metadata-eval 57.8%

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

   12. metadata-eval 57.8%

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

3. Simplified 57.8%

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

5. Taylor expanded in x around 0 81.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.16666666666666666 \cdot {x}^{2}\right)} \]
6. Step-by-step derivation

   1. metadata-eval 81.6%

      \[\leadsto x \cdot \left(1 + \color{blue}{\left(--0.16666666666666666\right)} \cdot {x}^{2}\right) \]

   2. cancel-sign-sub-inv 81.6%

      \[\leadsto x \cdot \color{blue}{\left(1 - -0.16666666666666666 \cdot {x}^{2}\right)} \]

   3. unpow2 81.6%

      \[\leadsto x \cdot \left(1 - -0.16666666666666666 \cdot \color{blue}{\left(x \cdot x\right)}\right) \]

   4. associate-*l* 81.6%

      \[\leadsto x \cdot \left(1 - \color{blue}{\left(-0.16666666666666666 \cdot x\right) \cdot x}\right) \]

   5. *-commutative 81.6%

      \[\leadsto x \cdot \left(1 - \color{blue}{x \cdot \left(-0.16666666666666666 \cdot x\right)}\right) \]

   6. *-commutative 81.6%

      \[\leadsto x \cdot \left(1 - \color{blue}{\left(-0.16666666666666666 \cdot x\right) \cdot x}\right) \]

   7. associate-*l* 81.6%

      \[\leadsto x \cdot \left(1 - \color{blue}{-0.16666666666666666 \cdot \left(x \cdot x\right)}\right) \]

   8. unpow2 81.6%

      \[\leadsto x \cdot \left(1 - -0.16666666666666666 \cdot \color{blue}{{x}^{2}}\right) \]

   9. cancel-sign-sub-inv 81.6%

      \[\leadsto x \cdot \color{blue}{\left(1 + \left(--0.16666666666666666\right) \cdot {x}^{2}\right)} \]

   10. metadata-eval 81.6%

       \[\leadsto x \cdot \left(1 + \color{blue}{0.16666666666666666} \cdot {x}^{2}\right) \]

   11. *-commutative 81.6%

       \[\leadsto x \cdot \left(1 + \color{blue}{{x}^{2} \cdot 0.16666666666666666}\right) \]

7. Simplified 81.6%

   \[\leadsto \color{blue}{x \cdot \left(1 + {x}^{2} \cdot 0.16666666666666666\right)} \]
8. Step-by-step derivation

   1. unpow2 89.5%

      \[\leadsto x \cdot \left(1 + {x}^{2} \cdot \left(0.16666666666666666 + \color{blue}{\left(x \cdot x\right)} \cdot 0.008333333333333333\right)\right) \]

9. Applied egg-rr 81.6%

   \[\leadsto x \cdot \left(1 + \color{blue}{\left(x \cdot x\right)} \cdot 0.16666666666666666\right) \]
10. Add Preprocessing

Alternative 5: 61.6% accurate, 29.4× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + x \cdot 0.25\right) \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* x (+ 1.0 (* x 0.25))))

C

double code(double x) {
	return x * (1.0 + (x * 0.25));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = x * (1.0d0 + (x * 0.25d0))
end function

Java

public static double code(double x) {
	return x * (1.0 + (x * 0.25));
}

Python

def code(x):
	return x * (1.0 + (x * 0.25))

Julia

function code(x)
	return Float64(x * Float64(1.0 + Float64(x * 0.25)))
end

MATLAB

function tmp = code(x)
	tmp = x * (1.0 + (x * 0.25));
end

Wolfram

code[x_] := N[(x * N[(1.0 + N[(x * 0.25), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.8%

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

3. Taylor expanded in x around 0 29.0%

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

   1. mul-1-neg 29.0%

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

   2. unsub-neg 29.0%

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

5. Simplified 29.0%

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

6. Taylor expanded in x around 0 58.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + 0.25 \cdot x\right)} \]
7. Step-by-step derivation

   1. *-commutative 58.7%

      \[\leadsto x \cdot \left(1 + \color{blue}{x \cdot 0.25}\right) \]

8. Simplified 58.7%

   \[\leadsto \color{blue}{x \cdot \left(1 + x \cdot 0.25\right)} \]
9. Add Preprocessing

Alternative 6: 52.0% 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 57.8%

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

   1. div-sub 57.8%

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

   2. *-rgt-identity 57.8%

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

   3. associate-*l/ 57.8%

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

   4. associate-/l* 57.8%

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

   5. fma-neg 57.8%

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

   6. metadata-eval 57.8%

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

   7. exp-neg 57.8%

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

   8. associate-/l/ 57.8%

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

   9. associate-/r* 57.8%

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

   10. distribute-neg-frac 57.8%

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

   11. metadata-eval 57.8%

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

   12. metadata-eval 57.8%

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

3. Simplified 57.8%

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

5. Taylor expanded in x around 0 48.7%

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

Reproduce

herbie shell --seed 2024129 
(FPCore (x)
  :name "Hyperbolic sine"
  :precision binary64
  (/ (- (exp x) (exp (- x))) 2.0))