Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 9.1s

Alternatives: 8

Speedup: 1.0×

• 50.0% 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

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

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: 71.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+151}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.8e-7)
   (/ (sin y) y)
   (if (<= x 8.5e+151) (cosh x) (* (sin y) (/ (+ 1.0 (* 0.5 (* x x))) y)))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.8e-7) {
		tmp = sin(y) / y;
	} else if (x <= 8.5e+151) {
		tmp = cosh(x);
	} else {
		tmp = sin(y) * ((1.0 + (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 <= 6.8d-7) then
        tmp = sin(y) / y
    else if (x <= 8.5d+151) then
        tmp = cosh(x)
    else
        tmp = sin(y) * ((1.0d0 + (0.5d0 * (x * x))) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.8e-7) {
		tmp = Math.sin(y) / y;
	} else if (x <= 8.5e+151) {
		tmp = Math.cosh(x);
	} else {
		tmp = Math.sin(y) * ((1.0 + (0.5 * (x * x))) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.8e-7:
		tmp = math.sin(y) / y
	elif x <= 8.5e+151:
		tmp = math.cosh(x)
	else:
		tmp = math.sin(y) * ((1.0 + (0.5 * (x * x))) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.8e-7)
		tmp = Float64(sin(y) / y);
	elseif (x <= 8.5e+151)
		tmp = cosh(x);
	else
		tmp = Float64(sin(y) * Float64(Float64(1.0 + Float64(0.5 * Float64(x * x))) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.8e-7)
		tmp = sin(y) / y;
	elseif (x <= 8.5e+151)
		tmp = cosh(x);
	else
		tmp = sin(y) * ((1.0 + (0.5 * (x * x))) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.8e-7], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 8.5e+151], N[Cosh[x], $MachinePrecision], N[(N[Sin[y], $MachinePrecision] * N[(N[(1.0 + N[(0.5 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.79999999999999948e-7

   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 66.6%

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

   if 6.79999999999999948e-7 < x < 8.50000000000000051e151

   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. inv-pow 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. div-inv 100.0%

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

      3. associate-/r* 100.0%

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

      4. clear-num 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0 68.8%

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

      1. associate-/l/ 68.8%

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

      2. cosh-def 68.8%

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

      3. lft-mult-inverse 68.8%

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

      4. associate-/l/ 68.8%

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

      5. metadata-eval 68.8%

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

      6. cosh-def 68.8%

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

      7. expm1-log1p-u 68.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x\right)\right)} \]

      8. expm1-udef 68.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]

   10. Applied egg-rr 68.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 68.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x\right)\right)} \]

       2. expm1-log1p 68.8%

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

   12. Simplified 68.8%

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

   if 8.50000000000000051e151 < 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 y around 0 100.0%

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

       1. unpow2 100.0%

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

   11. Simplified 100.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+151}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;\sin y \cdot \frac{1 + 0.5 \cdot \left(x \cdot x\right)}{y}\\ \end{array} \]

Alternative 3: 71.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+151}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 6.8e-7)
   (/ (sin y) y)
   (if (<= x 8.5e+151) (cosh x) (* 0.5 (/ (sin y) (/ y (* x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.8e-7) {
		tmp = sin(y) / y;
	} else if (x <= 8.5e+151) {
		tmp = cosh(x);
	} else {
		tmp = 0.5 * (sin(y) / (y / (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 <= 6.8d-7) then
        tmp = sin(y) / y
    else if (x <= 8.5d+151) then
        tmp = cosh(x)
    else
        tmp = 0.5d0 * (sin(y) / (y / (x * x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 6.8e-7) {
		tmp = Math.sin(y) / y;
	} else if (x <= 8.5e+151) {
		tmp = Math.cosh(x);
	} else {
		tmp = 0.5 * (Math.sin(y) / (y / (x * x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.8e-7:
		tmp = math.sin(y) / y
	elif x <= 8.5e+151:
		tmp = math.cosh(x)
	else:
		tmp = 0.5 * (math.sin(y) / (y / (x * x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.8e-7)
		tmp = Float64(sin(y) / y);
	elseif (x <= 8.5e+151)
		tmp = cosh(x);
	else
		tmp = Float64(0.5 * Float64(sin(y) / Float64(y / Float64(x * x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.8e-7)
		tmp = sin(y) / y;
	elseif (x <= 8.5e+151)
		tmp = cosh(x);
	else
		tmp = 0.5 * (sin(y) / (y / (x * x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.8e-7], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], If[LessEqual[x, 8.5e+151], N[Cosh[x], $MachinePrecision], N[(0.5 * N[(N[Sin[y], $MachinePrecision] / N[(y / N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < 6.79999999999999948e-7

   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 66.6%

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

   if 6.79999999999999948e-7 < x < 8.50000000000000051e151

   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. inv-pow 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. div-inv 100.0%

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

      3. associate-/r* 100.0%

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

      4. clear-num 100.0%

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

   7. Applied egg-rr 100.0%

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

   8. Taylor expanded in y around 0 68.8%

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

      1. associate-/l/ 68.8%

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

      2. cosh-def 68.8%

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

      3. lft-mult-inverse 68.8%

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

      4. associate-/l/ 68.8%

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

      5. metadata-eval 68.8%

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

      6. cosh-def 68.8%

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

      7. expm1-log1p-u 68.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x\right)\right)} \]

      8. expm1-udef 68.8%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]

   10. Applied egg-rr 68.8%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 68.8%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x\right)\right)} \]

       2. expm1-log1p 68.8%

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

   12. Simplified 68.8%

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

   if 8.50000000000000051e151 < 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%

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

       1. unpow2 100.0%

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

       2. associate-/l* 100.0%

          \[\leadsto 0.5 \cdot \color{blue}{\frac{\sin y}{\frac{y}{x \cdot x}}} \]

   11. Simplified 100.0%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{elif}\;x \leq 8.5 \cdot 10^{+151}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;0.5 \cdot \frac{\sin y}{\frac{y}{x \cdot x}}\\ \end{array} \]

Alternative 4: 68.5% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 6.8e-7) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 6.8e-7) {
		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 (x <= 6.8d-7) 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 (x <= 6.8e-7) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 6.8e-7:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 6.8e-7)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 6.8e-7)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 6.8e-7], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 6.79999999999999948e-7

   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 66.6%

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

   if 6.79999999999999948e-7 < 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. inv-pow 100.0%

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

   5. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

      2. div-inv 97.9%

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

      3. associate-/r* 97.9%

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

      4. clear-num 97.9%

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

   7. Applied egg-rr 97.9%

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

   8. Taylor expanded in y around 0 72.9%

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

      1. associate-/l/ 72.9%

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

      2. cosh-def 72.9%

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

      3. lft-mult-inverse 75.0%

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

      4. associate-/l/ 75.0%

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

      5. metadata-eval 75.0%

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

      6. cosh-def 75.0%

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

      7. expm1-log1p-u 75.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x\right)\right)} \]

      8. expm1-udef 75.0%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]

   10. Applied egg-rr 75.0%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 75.0%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x\right)\right)} \]

       2. expm1-log1p 75.0%

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

   12. Simplified 75.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 6.8 \cdot 10^{-7}:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 5: 62.6% 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. *-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 99.9%

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

   2. inv-pow 99.9%

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

5. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

   2. div-inv 99.4%

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

   3. associate-/r* 99.5%

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

   4. clear-num 99.5%

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

7. Applied egg-rr 99.5%

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

8. Taylor expanded in y around 0 66.0%

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

   1. associate-/l/ 65.9%

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

   2. cosh-def 65.9%

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

   3. lft-mult-inverse 66.4%

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

   4. associate-/l/ 66.4%

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

   5. metadata-eval 66.4%

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

   6. cosh-def 66.4%

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

   7. expm1-log1p-u 66.4%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x\right)\right)} \]

   8. expm1-udef 66.4%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]

10. Applied egg-rr 66.4%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 66.4%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x\right)\right)} \]

    2. expm1-log1p 66.4%

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

12. Simplified 66.4%

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

13. Final simplification 66.4%

    \[\leadsto \cosh x \]

Alternative 6: 50.9% accurate, 13.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(1.0 / y), $MachinePrecision] + N[(0.5 * N[(N[(x * x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 / y), $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 99.9%

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

   2. inv-pow 99.9%

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

5. Applied egg-rr 99.9%

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

   1. unpow-1 99.9%

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

   2. div-inv 99.4%

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

   3. associate-/r* 99.5%

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

   4. clear-num 99.5%

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

7. Applied egg-rr 99.5%

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

8. Taylor expanded in y around 0 66.0%

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

9. Taylor expanded in x around 0 57.9%

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

    1. unpow2 87.9%

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

11. Simplified 57.9%

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

12. Final simplification 57.9%

    \[\leadsto \frac{\frac{1}{y} + 0.5 \cdot \frac{x \cdot x}{y}}{\frac{1}{y}} \]

Alternative 7: 44.7% 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 99.9%

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

   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 87.9%

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

   1. unpow2 87.9%

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

8. Simplified 87.9%

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

9. Taylor expanded in y around 0 53.9%

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

    1. unpow2 53.9%

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

11. Simplified 53.9%

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

12. Final simplification 53.9%

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

Alternative 8: 25.9% 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 54.6%

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

5. Taylor expanded in y around 0 32.0%

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

6. Final simplification 32.0%

   \[\leadsto 1 \]

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 2023268 
(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)))