Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.2s

Alternatives: 9

Speedup: 1.0×

• 49.3% 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

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

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 2: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.002) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.002) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (cosh(x) <= 1.002d0) then
        tmp = sin(y) / y
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.cosh(x) <= 1.002) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.002:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.002)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 1.002)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.002], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.002

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 99.6%

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

   if 1.002 < (cosh.f64 x)

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y \cdot \cosh x}}} \]
   4. Step-by-step derivation

      1. inv-pow 100.0%

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

      2. add-sqr-sqrt 76.3%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}} \cdot \sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}}^{-1} \]

      3. unpow-prod-down 76.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}^{-1} \cdot {\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}^{-1}} \]

      4. associate-/r* 76.3%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{\frac{y}{\sin y}}{\cosh x}}}\right)}^{-1} \cdot {\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}^{-1} \]

      5. associate-/r* 76.3%

         \[\leadsto {\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{\frac{\frac{y}{\sin y}}{\cosh x}}}\right)}^{-1} \]

   5. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-1} \cdot {\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. pow-sqr 76.3%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{\left(2 \cdot -1\right)}} \]

      2. metadata-eval 76.3%

         \[\leadsto {\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{\color{blue}{-2}} \]

   7. Simplified 76.3%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-2}} \]

   8. Taylor expanded in y around 0 76.3%

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

      1. +-commutative 76.3%

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

      2. rec-exp 76.3%

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

      3. unpow2 76.3%

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

      4. rem-square-sqrt 76.3%

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

      5. cosh-def 76.3%

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

   10. Simplified 76.3%

       \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 3: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.058:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot 0.5\right)}{\frac{y}{\sin y}}\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+148}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(0.5 \cdot \frac{x \cdot x}{y} + \frac{1}{y}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 0.058)
   (/ (+ 1.0 (* x (* x 0.5))) (/ y (sin y)))
   (if (<= x 9.4e+148)
     (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))
     (* (sin y) (+ (* 0.5 (/ (* x x) y)) (/ 1.0 y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 0.058) {
		tmp = (1.0 + (x * (x * 0.5))) / (y / sin(y));
	} else if (x <= 9.4e+148) {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = sin(y) * ((0.5 * ((x * x) / y)) + (1.0 / 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 <= 0.058d0) then
        tmp = (1.0d0 + (x * (x * 0.5d0))) / (y / sin(y))
    else if (x <= 9.4d+148) then
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    else
        tmp = sin(y) * ((0.5d0 * ((x * x) / y)) + (1.0d0 / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 0.058) {
		tmp = (1.0 + (x * (x * 0.5))) / (y / Math.sin(y));
	} else if (x <= 9.4e+148) {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	} else {
		tmp = Math.sin(y) * ((0.5 * ((x * x) / y)) + (1.0 / y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 0.058:
		tmp = (1.0 + (x * (x * 0.5))) / (y / math.sin(y))
	elif x <= 9.4e+148:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	else:
		tmp = math.sin(y) * ((0.5 * ((x * x) / y)) + (1.0 / y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 0.058)
		tmp = Float64(Float64(1.0 + Float64(x * Float64(x * 0.5))) / Float64(y / sin(y)));
	elseif (x <= 9.4e+148)
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	else
		tmp = Float64(sin(y) * Float64(Float64(0.5 * Float64(Float64(x * x) / y)) + Float64(1.0 / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 0.058)
		tmp = (1.0 + (x * (x * 0.5))) / (y / sin(y));
	elseif (x <= 9.4e+148)
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	else
		tmp = sin(y) * ((0.5 * ((x * x) / y)) + (1.0 / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 0.058], N[(N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 9.4e+148], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(0.5 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(1.0 / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 0.0580000000000000029

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

      3. *-commutative 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y \cdot \cosh x}}} \]
   4. Step-by-step derivation

      1. associate-/r* 99.9%

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

      2. clear-num 99.9%

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

      3. associate-/r/ 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 86.5%

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

      1. unpow-1 86.5%

         \[\leadsto \left(\color{blue}{{y}^{-1}} + 0.5 \cdot \frac{{x}^{2}}{y}\right) \cdot \sin y \]

      2. +-commutative 86.5%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y} + {y}^{-1}\right)} \cdot \sin y \]

      3. unpow2 86.5%

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

      4. unpow-1 86.5%

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

   8. Simplified 86.5%

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

   9. Taylor expanded in y around inf 83.0%

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

       1. associate-/l* 82.9%

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

       2. unpow2 82.9%

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

       3. *-commutative 82.9%

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

       4. associate-*l* 82.9%

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

   11. Simplified 82.9%

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

   if 0.0580000000000000029 < x < 9.3999999999999994e148

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.1%

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

      1. unpow2 78.1%

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

   4. Simplified 78.1%

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

   if 9.3999999999999994e148 < x

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y \cdot \cosh x}}} \]
   4. Step-by-step derivation

      1. associate-/r* 100.0%

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

      2. clear-num 100.0%

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

      3. associate-/r/ 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 97.7%

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

      1. unpow-1 97.7%

         \[\leadsto \left(\color{blue}{{y}^{-1}} + 0.5 \cdot \frac{{x}^{2}}{y}\right) \cdot \sin y \]

      2. +-commutative 97.7%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y} + {y}^{-1}\right)} \cdot \sin y \]

      3. unpow2 97.7%

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

      4. unpow-1 97.7%

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

   8. Simplified 97.7%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.058:\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot 0.5\right)}{\frac{y}{\sin y}}\\ \mathbf{elif}\;x \leq 9.4 \cdot 10^{+148}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \left(0.5 \cdot \frac{x \cdot x}{y} + \frac{1}{y}\right)\\ \end{array} \]

Alternative 4: 84.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 0.054 \lor \neg \left(x \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot 0.5\right)}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (or (<= x 0.054) (not (<= x 1.9e+154)))
   (/ (+ 1.0 (* x (* x 0.5))) (/ y (sin y)))
   (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if ((x <= 0.054) || !(x <= 1.9e+154)) {
		tmp = (1.0 + (x * (x * 0.5))) / (y / sin(y));
	} else {
		tmp = cosh(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 <= 0.054d0) .or. (.not. (x <= 1.9d+154))) then
        tmp = (1.0d0 + (x * (x * 0.5d0))) / (y / sin(y))
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if ((x <= 0.054) || !(x <= 1.9e+154)) {
		tmp = (1.0 + (x * (x * 0.5))) / (y / Math.sin(y));
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if (x <= 0.054) or not (x <= 1.9e+154):
		tmp = (1.0 + (x * (x * 0.5))) / (y / math.sin(y))
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if ((x <= 0.054) || !(x <= 1.9e+154))
		tmp = Float64(Float64(1.0 + Float64(x * Float64(x * 0.5))) / Float64(y / sin(y)));
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((x <= 0.054) || ~((x <= 1.9e+154)))
		tmp = (1.0 + (x * (x * 0.5))) / (y / sin(y));
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[Or[LessEqual[x, 0.054], N[Not[LessEqual[x, 1.9e+154]], $MachinePrecision]], N[(N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y / N[Sin[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0539999999999999994 or 1.8999999999999999e154 < x

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

      3. *-commutative 99.9%

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

   3. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y \cdot \cosh x}}} \]
   4. Step-by-step derivation

      1. associate-/r* 99.9%

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

      2. clear-num 99.9%

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

      3. associate-/r/ 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 88.8%

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

      1. unpow-1 88.8%

         \[\leadsto \left(\color{blue}{{y}^{-1}} + 0.5 \cdot \frac{{x}^{2}}{y}\right) \cdot \sin y \]

      2. +-commutative 88.8%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y} + {y}^{-1}\right)} \cdot \sin y \]

      3. unpow2 88.8%

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

      4. unpow-1 88.8%

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

   8. Simplified 88.8%

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

   9. Taylor expanded in y around inf 85.9%

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

       1. associate-/l* 85.9%

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

       2. unpow2 85.9%

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

       3. *-commutative 85.9%

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

       4. associate-*l* 85.9%

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

   11. Simplified 85.9%

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

   if 0.0539999999999999994 < x < 1.8999999999999999e154

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y \cdot \cosh x}}} \]
   4. Step-by-step derivation

      1. inv-pow 100.0%

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

      2. add-sqr-sqrt 82.4%

         \[\leadsto {\color{blue}{\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}} \cdot \sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}}^{-1} \]

      3. unpow-prod-down 82.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}^{-1} \cdot {\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}^{-1}} \]

      4. associate-/r* 82.4%

         \[\leadsto {\left(\sqrt{\color{blue}{\frac{\frac{y}{\sin y}}{\cosh x}}}\right)}^{-1} \cdot {\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}^{-1} \]

      5. associate-/r* 82.4%

         \[\leadsto {\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{\frac{\frac{y}{\sin y}}{\cosh x}}}\right)}^{-1} \]

   5. Applied egg-rr 82.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-1} \cdot {\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-1}} \]
   6. Step-by-step derivation

      1. pow-sqr 82.4%

         \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{\left(2 \cdot -1\right)}} \]

      2. metadata-eval 82.4%

         \[\leadsto {\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{\color{blue}{-2}} \]

   7. Simplified 82.4%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-2}} \]

   8. Taylor expanded in y around 0 82.4%

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

      1. +-commutative 82.4%

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

      2. rec-exp 82.4%

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

      3. unpow2 82.4%

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

      4. rem-square-sqrt 82.4%

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

      5. cosh-def 82.4%

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

   10. Simplified 82.4%

       \[\leadsto \color{blue}{\cosh x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.054 \lor \neg \left(x \leq 1.9 \cdot 10^{+154}\right):\\ \;\;\;\;\frac{1 + x \cdot \left(x \cdot 0.5\right)}{\frac{y}{\sin y}}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 5: 68.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 9e-15)
   (/ (sin y) y)
   (* (cosh x) (+ 1.0 (* -0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 9e-15) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * 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 <= 9d-15) then
        tmp = sin(y) / y
    else
        tmp = cosh(x) * (1.0d0 + ((-0.16666666666666666d0) * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 9e-15) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 9e-15:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 9e-15)
		tmp = Float64(sin(y) / y);
	else
		tmp = Float64(cosh(x) * Float64(1.0 + Float64(-0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 9e-15)
		tmp = sin(y) / y;
	else
		tmp = cosh(x) * (1.0 + (-0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 9e-15], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[(N[Cosh[x], $MachinePrecision] * N[(1.0 + N[(-0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.9999999999999995e-15

   1. Initial program 99.9%

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

   2. Taylor expanded in x around 0 66.4%

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

   if 8.9999999999999995e-15 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 78.1%

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

      1. unpow2 78.1%

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

   4. Simplified 78.1%

      \[\leadsto \cosh x \cdot \color{blue}{\left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 9 \cdot 10^{-15}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x \cdot \left(1 + -0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 6: 62.9% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y)
	return cosh(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Cosh[x], $MachinePrecision]

Derivation

1. Initial program 99.9%

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

   1. associate-*r/ 99.9%

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

   2. clear-num 99.9%

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

   3. *-commutative 99.9%

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

3. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y \cdot \cosh x}}} \]
4. Step-by-step derivation

   1. inv-pow 99.9%

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

   2. add-sqr-sqrt 73.7%

      \[\leadsto {\color{blue}{\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}} \cdot \sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}}^{-1} \]

   3. unpow-prod-down 73.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}^{-1} \cdot {\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}^{-1}} \]

   4. associate-/r* 73.7%

      \[\leadsto {\left(\sqrt{\color{blue}{\frac{\frac{y}{\sin y}}{\cosh x}}}\right)}^{-1} \cdot {\left(\sqrt{\frac{y}{\sin y \cdot \cosh x}}\right)}^{-1} \]

   5. associate-/r* 73.7%

      \[\leadsto {\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-1} \cdot {\left(\sqrt{\color{blue}{\frac{\frac{y}{\sin y}}{\cosh x}}}\right)}^{-1} \]

5. Applied egg-rr 73.7%

   \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-1} \cdot {\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-1}} \]
6. Step-by-step derivation

   1. pow-sqr 73.7%

      \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{\left(2 \cdot -1\right)}} \]

   2. metadata-eval 73.7%

      \[\leadsto {\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{\color{blue}{-2}} \]

7. Simplified 73.7%

   \[\leadsto \color{blue}{{\left(\sqrt{\frac{\frac{y}{\sin y}}{\cosh x}}\right)}^{-2}} \]

8. Taylor expanded in y around 0 61.6%

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

   1. +-commutative 61.6%

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

   2. rec-exp 61.6%

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

   3. unpow2 61.6%

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

   4. rem-square-sqrt 62.1%

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

   5. cosh-def 62.1%

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

10. Simplified 62.1%

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

11. Final simplification 62.1%

    \[\leadsto \cosh x \]

Alternative 7: 39.5% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.75 \cdot 10^{+157}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 2.75e+157)
   (+ 1.0 (* y (* y -0.16666666666666666)))
   (+ 1.0 (* x (* x 0.5)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 2.75e+157) {
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	} else {
		tmp = 1.0 + (x * (x * 0.5));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 2.75d+157) then
        tmp = 1.0d0 + (y * (y * (-0.16666666666666666d0)))
    else
        tmp = 1.0d0 + (x * (x * 0.5d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 2.75e+157) {
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	} else {
		tmp = 1.0 + (x * (x * 0.5));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 2.75e+157:
		tmp = 1.0 + (y * (y * -0.16666666666666666))
	else:
		tmp = 1.0 + (x * (x * 0.5))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 2.75e+157)
		tmp = Float64(1.0 + Float64(y * Float64(y * -0.16666666666666666)));
	else
		tmp = Float64(1.0 + Float64(x * Float64(x * 0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 2.75e+157)
		tmp = 1.0 + (y * (y * -0.16666666666666666));
	else
		tmp = 1.0 + (x * (x * 0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 2.75e+157], N[(1.0 + N[(y * N[(y * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(x * N[(x * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7500000000000001e157

   1. Initial program 99.9%

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

      1. associate-*r/ 99.9%

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

      2. clear-num 99.9%

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

      3. *-commutative 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in x around 0 56.6%

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

   5. Taylor expanded in y around 0 30.7%

      \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}} \]
   6. Step-by-step derivation

      1. *-commutative 30.7%

         \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

      2. unpow2 30.7%

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

   7. Simplified 30.7%

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

   8. Taylor expanded in y around 0 30.7%

      \[\leadsto 1 + \color{blue}{-0.16666666666666666 \cdot {y}^{2}} \]
   9. Step-by-step derivation

      1. *-commutative 30.7%

         \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

      2. unpow2 30.7%

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

      3. associate-*r* 30.7%

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

   10. Simplified 30.7%

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

   if 2.7500000000000001e157 < x

   1. Initial program 100.0%

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

      1. associate-*r/ 100.0%

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

      2. clear-num 100.0%

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

      3. *-commutative 100.0%

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

   3. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y \cdot \cosh x}}} \]
   4. Step-by-step derivation

      1. associate-/r* 100.0%

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

      2. clear-num 100.0%

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

      3. associate-/r/ 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. unpow-1 100.0%

         \[\leadsto \left(\color{blue}{{y}^{-1}} + 0.5 \cdot \frac{{x}^{2}}{y}\right) \cdot \sin y \]

      2. +-commutative 100.0%

         \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y} + {y}^{-1}\right)} \cdot \sin y \]

      3. unpow2 100.0%

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

      4. unpow-1 100.0%

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

   8. Simplified 100.0%

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

   9. Taylor expanded in y around 0 68.4%

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

       1. unpow2 68.4%

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

       2. *-commutative 68.4%

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

       3. associate-*l* 68.4%

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

   11. Simplified 68.4%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.75 \cdot 10^{+157}:\\ \;\;\;\;1 + y \cdot \left(y \cdot -0.16666666666666666\right)\\ \mathbf{else}:\\ \;\;\;\;1 + x \cdot \left(x \cdot 0.5\right)\\ \end{array} \]

Alternative 8: 45.1% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. associate-*r/ 99.9%

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

   2. clear-num 99.9%

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

   3. *-commutative 99.9%

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

3. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{1}{\frac{y}{\sin y \cdot \cosh x}}} \]
4. Step-by-step derivation

   1. associate-/r* 99.9%

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

   2. clear-num 99.9%

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

   3. associate-/r/ 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 80.9%

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

   1. unpow-1 80.9%

      \[\leadsto \left(\color{blue}{{y}^{-1}} + 0.5 \cdot \frac{{x}^{2}}{y}\right) \cdot \sin y \]

   2. +-commutative 80.9%

      \[\leadsto \color{blue}{\left(0.5 \cdot \frac{{x}^{2}}{y} + {y}^{-1}\right)} \cdot \sin y \]

   3. unpow2 80.9%

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

   4. unpow-1 80.9%

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

8. Simplified 80.9%

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

9. Taylor expanded in y around 0 42.7%

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

    1. unpow2 42.7%

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

    2. *-commutative 42.7%

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

    3. associate-*l* 42.7%

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

11. Simplified 42.7%

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

12. Final simplification 42.7%

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

Alternative 9: 26.5% 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. associate-*r/ 99.9%

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

   2. clear-num 99.9%

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

   3. *-commutative 99.9%

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

3. Applied egg-rr 99.9%

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

4. Taylor expanded in x around 0 48.6%

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

5. Taylor expanded in y around 0 23.3%

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

6. Final simplification 23.3%

   \[\leadsto 1 \]

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 2023215 
(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)))