Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 9.8s

Alternatives: 11

Speedup: 1.0×

• 49.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \cosh x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

double code(double x, double y) {
	return cosh(x) * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cosh(x) * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cosh(x) * (Math.sin(y) / y);
}

Python

def code(x, y):
	return math.cosh(x) * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(cosh(x) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cosh(x) * (sin(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Initial Program: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

double code(double x, double y) {
	return cosh(x) * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cosh(x) * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cosh(x) * (Math.sin(y) / y);
}

Python

def code(x, y):
	return math.cosh(x) * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(cosh(x) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cosh(x) * (sin(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\sin y}{\frac{y}{\cosh x}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (sin y) (/ y (cosh x))))

C

double code(double x, double y) {
	return sin(y) / (y / cosh(x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(y) / (y / cosh(x))
end function

Java

public static double code(double x, double y) {
	return Math.sin(y) / (y / Math.cosh(x));
}

Python

def code(x, y):
	return math.sin(y) / (y / math.cosh(x))

Julia

function code(x, y)
	return Float64(sin(y) / Float64(y / cosh(x)))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(y) / (y / cosh(x));
end

Wolfram

code[x_, y_] := N[(N[Sin[y], $MachinePrecision] / N[(y / N[Cosh[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

   2. associate-/r/ 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

4. Final simplification 99.9%

   \[\leadsto \frac{\sin y}{\frac{y}{\cosh x}} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \cosh x \cdot \frac{\sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* (cosh x) (/ (sin y) y)))

C

double code(double x, double y) {
	return cosh(x) * (sin(y) / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = cosh(x) * (sin(y) / y)
end function

Java

public static double code(double x, double y) {
	return Math.cosh(x) * (Math.sin(y) / y);
}

Python

def code(x, y):
	return math.cosh(x) * (math.sin(y) / y)

Julia

function code(x, y)
	return Float64(cosh(x) * Float64(sin(y) / y))
end

MATLAB

function tmp = code(x, y)
	tmp = cosh(x) * (sin(y) / y);
end

Wolfram

code[x_, y_] := N[(N[Cosh[x], $MachinePrecision] * N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]

2. Final simplification 99.9%

   \[\leadsto \cosh x \cdot \frac{\sin y}{y} \]

Alternative 3: 66.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+133}:\\ \;\;\;\;y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(y \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7e-10)
   (/ (sin y) y)
   (if (<= x 3e+133)
     (* y (+ (/ 1.0 y) (* 0.5 (/ x (/ y x)))))
     (* (sin y) (* y (* 0.5 (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7e-10) {
		tmp = sin(y) / y;
	} else if (x <= 3e+133) {
		tmp = y * ((1.0 / y) + (0.5 * (x / (y / x))));
	} else {
		tmp = sin(y) * (y * (0.5 * (x * x)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 7d-10) then
        tmp = sin(y) / y
    else if (x <= 3d+133) then
        tmp = y * ((1.0d0 / y) + (0.5d0 * (x / (y / x))))
    else
        tmp = sin(y) * (y * (0.5d0 * (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7e-10) {
		tmp = Math.sin(y) / y;
	} else if (x <= 3e+133) {
		tmp = y * ((1.0 / y) + (0.5 * (x / (y / x))));
	} else {
		tmp = Math.sin(y) * (y * (0.5 * (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7e-10:
		tmp = math.sin(y) / y
	elif x <= 3e+133:
		tmp = y * ((1.0 / y) + (0.5 * (x / (y / x))))
	else:
		tmp = math.sin(y) * (y * (0.5 * (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7e-10)
		tmp = Float64(sin(y) / y);
	elseif (x <= 3e+133)
		tmp = Float64(y * Float64(Float64(1.0 / y) + Float64(0.5 * Float64(x / Float64(y / x)))));
	else
		tmp = Float64(sin(y) * Float64(y * Float64(0.5 * Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7e-10)
		tmp = sin(y) / y;
	elseif (x <= 3e+133)
		tmp = y * ((1.0 / y) + (0.5 * (x / (y / x))));
	else
		tmp = sin(y) * (y * (0.5 * (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7e-10], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 3e+133], N[(y * N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(y * N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.99999999999999961e-10

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   4. Taylor expanded in x around 0 67.5%

      \[\leadsto \color{blue}{\frac{\sin y}{y}} \]

   if 6.99999999999999961e-10 < x < 3.00000000000000007e133

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

      3. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   6. Taylor expanded in x around 0 29.3%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 29.3%

         \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   8. Simplified 29.3%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

   9. Taylor expanded in x around 0 29.3%

      \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \color{blue}{\frac{{x}^{2}}{y}}\right) \cdot \sin y \]
   10. Step-by-step derivation

       1. unpow2 29.3%

          \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

       2. associate-/l* 29.3%

          \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \color{blue}{\frac{x}{\frac{y}{x}}}\right) \cdot \sin y \]

   11. Simplified 29.3%

       \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \color{blue}{\frac{x}{\frac{y}{x}}}\right) \cdot \sin y \]

   12. Taylor expanded in y around 0 29.1%

       \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{x}{\frac{y}{x}}\right) \cdot \color{blue}{y} \]

   if 3.00000000000000007e133 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

      3. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   6. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 100.0%

         \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   8. Simplified 100.0%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

   9. Taylor expanded in x around inf 100.0%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   10. Step-by-step derivation

       1. unpow2 100.0%

          \[\leadsto \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   11. Simplified 100.0%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]
   12. Step-by-step derivation

       1. expm1-log1p-u 81.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\left(0.5 \cdot \frac{x \cdot x}{y}\right) \cdot \sin y\right)\right)} \]

       2. expm1-udef 81.3%

          \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\left(0.5 \cdot \frac{x \cdot x}{y}\right) \cdot \sin y\right)} - 1} \]

   13. Applied egg-rr 81.3%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\sin y \cdot \left(\left(x \cdot x\right) \cdot \left(y \cdot 0.5\right)\right)\right)} - 1} \]
   14. Step-by-step derivation

       1. expm1-def 81.3%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin y \cdot \left(\left(x \cdot x\right) \cdot \left(y \cdot 0.5\right)\right)\right)\right)} \]

       2. expm1-log1p 100.0%

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

       3. *-commutative 100.0%

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

       4. associate-*r* 100.0%

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

   15. Simplified 100.0%

       \[\leadsto \color{blue}{\sin y \cdot \left(y \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 3 \cdot 10^{+133}:\\ \;\;\;\;y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x}{\frac{y}{x}}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(y \cdot \left(0.5 \cdot \left(x \cdot x\right)\right)\right)\\ \end{array} \]

Alternative 4: 78.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 500000:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(0.5 \cdot \frac{x \cdot x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 500000.0)
   (/ (+ 1.0 (* 0.5 (* x x))) (/ y (sin y)))
   (* (sin y) (* 0.5 (/ (* x x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 500000.0) {
		tmp = (1.0 + (0.5 * (x * x))) / (y / sin(y));
	} else {
		tmp = sin(y) * (0.5 * ((x * x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 500000.0d0) then
        tmp = (1.0d0 + (0.5d0 * (x * x))) / (y / sin(y))
    else
        tmp = sin(y) * (0.5d0 * ((x * x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 500000.0) {
		tmp = (1.0 + (0.5 * (x * x))) / (y / Math.sin(y));
	} else {
		tmp = Math.sin(y) * (0.5 * ((x * x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 500000.0:
		tmp = (1.0 + (0.5 * (x * x))) / (y / math.sin(y))
	else:
		tmp = math.sin(y) * (0.5 * ((x * x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 500000.0)
		tmp = Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) / Float64(y / sin(y)));
	else
		tmp = Float64(sin(y) * Float64(0.5 * Float64(Float64(x * x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 500000.0)
		tmp = (1.0 + (0.5 * (x * x))) / (y / sin(y));
	else
		tmp = sin(y) * (0.5 * ((x * x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 500000.0], N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(0.5 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5e5

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
   4. Step-by-step derivation

      1. clear-num 99.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

      2. associate-/r/ 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

      3. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   6. Taylor expanded in x around 0 87.5%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 87.5%

         \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   8. Simplified 87.5%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

   9. Taylor expanded in y around inf 83.9%

      \[\leadsto \color{blue}{\frac{\left(1 + 0.5 \cdot {x}^{2}\right) \cdot \sin y}{y}} \]
   10. Step-by-step derivation

       1. associate-/l* 83.7%

          \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot {x}^{2}}{\frac{y}{\sin y}}} \]

       2. unpow2 83.7%

          \[\leadsto \frac{1 + 0.5 \cdot \color{blue}{\left(x \cdot x\right)}}{\frac{y}{\sin y}} \]

   11. Simplified 83.7%

       \[\leadsto \color{blue}{\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{\frac{y}{\sin y}}} \]

   if 5e5 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

      3. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   6. Taylor expanded in x around 0 67.9%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 67.9%

         \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   8. Simplified 67.9%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

   9. Taylor expanded in x around inf 67.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   10. Step-by-step derivation

       1. unpow2 67.9%

          \[\leadsto \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   11. Simplified 67.9%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 500000:\\ \;\;\;\;\frac{1 + 0.5 \cdot \left(x \cdot x\right)}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(0.5 \cdot \frac{x \cdot x}{y}\right)\\ \end{array} \]

Alternative 5: 81.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \sin y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* (sin y) (+ (/ 1.0 y) (* 0.5 (/ (* x x) y)))))

C

double code(double x, double y) {
	return sin(y) * ((1.0 / y) + (0.5 * ((x * x) / y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = sin(y) * ((1.0d0 / y) + (0.5d0 * ((x * x) / y)))
end function

Java

public static double code(double x, double y) {
	return Math.sin(y) * ((1.0 / y) + (0.5 * ((x * x) / y)));
}

Python

def code(x, y):
	return math.sin(y) * ((1.0 / y) + (0.5 * ((x * x) / y)))

Julia

function code(x, y)
	return Float64(sin(y) * Float64(Float64(1.0 / y) + Float64(0.5 * Float64(Float64(x * x) / y))))
end

MATLAB

function tmp = code(x, y)
	tmp = sin(y) * ((1.0 / y) + (0.5 * ((x * x) / y)));
end

Wolfram

code[x_, y_] := N[(N[Sin[y], $MachinePrecision] * N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

   2. associate-/r/ 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
4. Step-by-step derivation

   1. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

   2. associate-/r/ 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

   3. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

6. Taylor expanded in x around 0 83.1%

   \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
7. Step-by-step derivation

   1. unpow2 83.1%

      \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

8. Simplified 83.1%

   \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

9. Final simplification 83.1%

   \[\leadsto \sin y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right) \]

Alternative 6: 66.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(0.5 \cdot \frac{x \cdot x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1.4) (/ (sin y) y) (* (sin y) (* 0.5 (/ (* x x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1.4) {
		tmp = sin(y) / y;
	} else {
		tmp = sin(y) * (0.5 * ((x * x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1.4d0) then
        tmp = sin(y) / y
    else
        tmp = sin(y) * (0.5d0 * ((x * x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1.4) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.sin(y) * (0.5 * ((x * x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1.4:
		tmp = math.sin(y) / y
	else:
		tmp = math.sin(y) * (0.5 * ((x * x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1.4)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(sin(y) * Float64(0.5 * Float64(Float64(x * x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1.4)
		tmp = sin(y) / y;
	else
		tmp = sin(y) * (0.5 * ((x * x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1.4], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(0.5 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3999999999999999

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   4. Taylor expanded in x around 0 67.5%

      \[\leadsto \color{blue}{\frac{\sin y}{y}} \]

   if 1.3999999999999999 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

      3. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   6. Taylor expanded in x around 0 65.6%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 65.6%

         \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

   9. Taylor expanded in x around inf 65.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   10. Step-by-step derivation

       1. unpow2 65.6%

          \[\leadsto \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   11. Simplified 65.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]
3. Recombined 2 regimes into one program.

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.4:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(0.5 \cdot \frac{x \cdot x}{y}\right)\\ \end{array} \]

Alternative 7: 63.1% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x}{\frac{y}{x}}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 7e-10) (/ (sin y) y) (* y (+ (/ 1.0 y) (* 0.5 (/ x (/ y x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 7e-10) {
		tmp = sin(y) / y;
	} else {
		tmp = y * ((1.0 / y) + (0.5 * (x / (y / x))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 7d-10) then
        tmp = sin(y) / y
    else
        tmp = y * ((1.0d0 / y) + (0.5d0 * (x / (y / x))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 7e-10) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = y * ((1.0 / y) + (0.5 * (x / (y / x))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 7e-10:
		tmp = math.sin(y) / y
	else:
		tmp = y * ((1.0 / y) + (0.5 * (x / (y / x))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 7e-10)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(y * Float64(Float64(1.0 / y) + Float64(0.5 * Float64(x / Float64(y / x)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 7e-10)
		tmp = sin(y) / y;
	else
		tmp = y * ((1.0 / y) + (0.5 * (x / (y / x))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 7e-10], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(y * N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.99999999999999961e-10

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   4. Taylor expanded in x around 0 67.5%

      \[\leadsto \color{blue}{\frac{\sin y}{y}} \]

   if 6.99999999999999961e-10 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

      3. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   6. Taylor expanded in x around 0 66.4%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 66.4%

         \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   8. Simplified 66.4%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

   9. Taylor expanded in x around 0 66.4%

      \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \color{blue}{\frac{{x}^{2}}{y}}\right) \cdot \sin y \]
   10. Step-by-step derivation

       1. unpow2 66.4%

          \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

       2. associate-/l* 55.4%

          \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \color{blue}{\frac{x}{\frac{y}{x}}}\right) \cdot \sin y \]

   11. Simplified 55.4%

       \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \color{blue}{\frac{x}{\frac{y}{x}}}\right) \cdot \sin y \]

   12. Taylor expanded in y around 0 56.5%

       \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{x}{\frac{y}{x}}\right) \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 64.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7 \cdot 10^{-10}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x}{\frac{y}{x}}\right)\\ \end{array} \]

Alternative 8: 51.0% accurate, 15.8× speedup

Math

\[\begin{array}{l} \\ y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x}{\frac{y}{x}}\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (+ (/ 1.0 y) (* 0.5 (/ x (/ y x))))))

C

double code(double x, double y) {
	return y * ((1.0 / y) + (0.5 * (x / (y / x))));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = y * ((1.0d0 / y) + (0.5d0 * (x / (y / x))))
end function

Java

public static double code(double x, double y) {
	return y * ((1.0 / y) + (0.5 * (x / (y / x))));
}

Python

def code(x, y):
	return y * ((1.0 / y) + (0.5 * (x / (y / x))))

Julia

function code(x, y)
	return Float64(y * Float64(Float64(1.0 / y) + Float64(0.5 * Float64(x / Float64(y / x)))))
end

MATLAB

function tmp = code(x, y)
	tmp = y * ((1.0 / y) + (0.5 * (x / (y / x))));
end

Wolfram

code[x_, y_] := N[(y * N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(x / N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

   2. associate-/r/ 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
4. Step-by-step derivation

   1. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

   2. associate-/r/ 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

   3. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

6. Taylor expanded in x around 0 83.1%

   \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
7. Step-by-step derivation

   1. unpow2 83.1%

      \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

8. Simplified 83.1%

   \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

9. Taylor expanded in x around 0 83.1%

   \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \color{blue}{\frac{{x}^{2}}{y}}\right) \cdot \sin y \]
10. Step-by-step derivation

    1. unpow2 83.1%

       \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

    2. associate-/l* 76.9%

       \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \color{blue}{\frac{x}{\frac{y}{x}}}\right) \cdot \sin y \]

11. Simplified 76.9%

    \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \color{blue}{\frac{x}{\frac{y}{x}}}\right) \cdot \sin y \]

12. Taylor expanded in y around 0 52.7%

    \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{x}{\frac{y}{x}}\right) \cdot \color{blue}{y} \]

13. Final simplification 52.7%

    \[\leadsto y \cdot \left(\frac{1}{y} + 0.5 \cdot \frac{x}{\frac{y}{x}}\right) \]

Alternative 9: 48.1% accurate, 18.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 235:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x \cdot x}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 235.0) (+ 1.0 (* 0.5 (* x x))) (* y (* 0.5 (/ (* x x) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 235.0) {
		tmp = 1.0 + (0.5 * (x * x));
	} else {
		tmp = y * (0.5 * ((x * x) / y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 235.0d0) then
        tmp = 1.0d0 + (0.5d0 * (x * x))
    else
        tmp = y * (0.5d0 * ((x * x) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 235.0) {
		tmp = 1.0 + (0.5 * (x * x));
	} else {
		tmp = y * (0.5 * ((x * x) / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 235.0:
		tmp = 1.0 + (0.5 * (x * x))
	else:
		tmp = y * (0.5 * ((x * x) / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 235.0)
		tmp = Float64(1.0 + Float64(0.5 * Float64(x * x)));
	else
		tmp = Float64(y * Float64(0.5 * Float64(Float64(x * x) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 235.0)
		tmp = 1.0 + (0.5 * (x * x));
	else
		tmp = y * (0.5 * ((x * x) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 235.0], N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(0.5 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 235

   1. Initial program 99.9%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 99.9%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
   4. Step-by-step derivation

      1. clear-num 99.8%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

      2. associate-/r/ 99.7%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

      3. clear-num 99.7%

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

   5. Applied egg-rr 99.7%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   6. Taylor expanded in x around 0 88.4%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 88.4%

         \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   8. Simplified 88.4%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

   9. Taylor expanded in y around 0 48.1%

      \[\leadsto \color{blue}{1 + 0.5 \cdot {x}^{2}} \]
   10. Step-by-step derivation

       1. unpow2 48.1%

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

   11. Simplified 48.1%

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

   if 235 < x

   1. Initial program 100.0%

      \[\cosh x \cdot \frac{\sin y}{y} \]
   2. Step-by-step derivation

      1. *-commutative 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
   4. Step-by-step derivation

      1. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

      2. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

      3. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

   6. Taylor expanded in x around 0 65.6%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   7. Step-by-step derivation

      1. unpow2 65.6%

         \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   8. Simplified 65.6%

      \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

   9. Taylor expanded in x around inf 65.6%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
   10. Step-by-step derivation

       1. unpow2 65.6%

          \[\leadsto \left(0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

   11. Simplified 65.6%

       \[\leadsto \color{blue}{\left(0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

   12. Taylor expanded in y around 0 55.4%

       \[\leadsto \left(0.5 \cdot \frac{x \cdot x}{y}\right) \cdot \color{blue}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 49.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 235:\\ \;\;\;\;1 + 0.5 \cdot \left(x \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(0.5 \cdot \frac{x \cdot x}{y}\right)\\ \end{array} \]

Alternative 10: 44.9% accurate, 29.3× speedup

Math

\[\begin{array}{l} \\ 1 + 0.5 \cdot \left(x \cdot x\right) \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ 1.0 (* 0.5 (* x x))))

C

double code(double x, double y) {
	return 1.0 + (0.5 * (x * x));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0 + (0.5d0 * (x * x))
end function

Java

public static double code(double x, double y) {
	return 1.0 + (0.5 * (x * x));
}

Python

def code(x, y):
	return 1.0 + (0.5 * (x * x))

Julia

function code(x, y)
	return Float64(1.0 + Float64(0.5 * Float64(x * x)))
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0 + (0.5 * (x * x));
end

Wolfram

code[x_, y_] := N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

   2. associate-/r/ 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]
4. Step-by-step derivation

   1. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{\frac{y}{\cosh x}}{\sin y}}} \]

   2. associate-/r/ 99.8%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\cosh x}} \cdot \sin y} \]

   3. clear-num 99.8%

      \[\leadsto \color{blue}{\frac{\cosh x}{y}} \cdot \sin y \]

5. Applied egg-rr 99.8%

   \[\leadsto \color{blue}{\frac{\cosh x}{y} \cdot \sin y} \]

6. Taylor expanded in x around 0 83.1%

   \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{{x}^{2}}{y}\right)} \cdot \sin y \]
7. Step-by-step derivation

   1. unpow2 83.1%

      \[\leadsto \left(\frac{1}{y} + 0.5 \cdot \frac{\color{blue}{x \cdot x}}{y}\right) \cdot \sin y \]

8. Simplified 83.1%

   \[\leadsto \color{blue}{\left(\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}\right)} \cdot \sin y \]

9. Taylor expanded in y around 0 47.6%

   \[\leadsto \color{blue}{1 + 0.5 \cdot {x}^{2}} \]
10. Step-by-step derivation

    1. unpow2 47.6%

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

11. Simplified 47.6%

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

12. Final simplification 47.6%

    \[\leadsto 1 + 0.5 \cdot \left(x \cdot x\right) \]

Alternative 11: 27.1% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 1.0)

C

double code(double x, double y) {
	return 1.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 1.0d0
end function

Java

public static double code(double x, double y) {
	return 1.0;
}

Python

def code(x, y):
	return 1.0

Julia

function code(x, y)
	return 1.0
end

MATLAB

function tmp = code(x, y)
	tmp = 1.0;
end

Wolfram

code[x_, y_] := 1.0

Derivation

1. Initial program 99.9%

   \[\cosh x \cdot \frac{\sin y}{y} \]
2. Step-by-step derivation

   1. *-commutative 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{y} \cdot \cosh x} \]

   2. associate-/r/ 99.9%

      \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{\sin y}{\frac{y}{\cosh x}}} \]

4. Taylor expanded in x around 0 52.6%

   \[\leadsto \color{blue}{\frac{\sin y}{y}} \]

5. Taylor expanded in y around 0 27.9%

   \[\leadsto \color{blue}{1} \]

6. Final simplification 27.9%

   \[\leadsto 1 \]

Developer target: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\cosh x \cdot \sin y}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (* (cosh x) (sin y)) y))

C

double code(double x, double y) {
	return (cosh(x) * sin(y)) / y;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (cosh(x) * sin(y)) / y
end function

Java

public static double code(double x, double y) {
	return (Math.cosh(x) * Math.sin(y)) / y;
}

Python

def code(x, y):
	return (math.cosh(x) * math.sin(y)) / y

Julia

function code(x, y)
	return Float64(Float64(cosh(x) * sin(y)) / y)
end

MATLAB

function tmp = code(x, y)
	tmp = (cosh(x) * sin(y)) / y;
end

Wolfram

code[x_, y_] := N[(N[(N[Cosh[x], $MachinePrecision] * N[Sin[y], $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]

Reproduce

herbie shell --seed 2023279 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (/ (* (cosh x) (sin y)) y)

  (* (cosh x) (/ (sin y) y)))