Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.4% → 99.8%

Time: 6.8s

Alternatives: 8

Speedup: 1.0×

• 50.9% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing
5. Add Preprocessing

Alternative 2: 62.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{-16}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2e-16) (/ x (/ x y)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2e-16) {
		tmp = x / (x / y);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (sinh(y) <= 2d-16) then
        tmp = x / (x / y)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 2e-16) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2e-16:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2e-16)
		tmp = Float64(x / Float64(x / y));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2e-16)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-16], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2e-16

   1. Initial program 86.3%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 45.6%

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

      1. associate-/l* 59.2%

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

   7. Simplified 59.2%

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

   8. Taylor expanded in y around 0 45.6%

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

      1. associate-*l/ 69.7%

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

      2. *-commutative 69.7%

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

   10. Simplified 69.7%

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

   11. Taylor expanded in x around 0 57.4%

       \[\leadsto \color{blue}{x} \cdot \frac{y}{x} \]
   12. Step-by-step derivation

       1. clear-num 58.1%

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

       2. div-inv 57.7%

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

   13. Applied egg-rr 57.7%

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

   if 2e-16 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 76.0%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]
   6. Step-by-step derivation

      1. clear-num 76.0%

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

      2. un-div-inv 76.0%

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

   7. Applied egg-rr 76.0%

      \[\leadsto \color{blue}{\frac{x}{\frac{x}{\sinh y}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 76.0%

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

      2. *-inverses 76.0%

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

      3. *-lft-identity 76.0%

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

   9. Simplified 76.0%

      \[\leadsto \color{blue}{\sinh y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 68.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0045:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{elif}\;y \leq 3.4 \cdot 10^{+264}:\\ \;\;\;\;x \cdot \frac{\sinh y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{\frac{x}{y}}{x + -0.16666666666666666 \cdot {x}^{3}}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0045)
   (* y (/ (sin x) x))
   (if (<= y 3.4e+264)
     (* x (/ (sinh y) x))
     (/ 1.0 (/ (/ x y) (+ x (* -0.16666666666666666 (pow x 3.0))))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0045) {
		tmp = y * (sin(x) / x);
	} else if (y <= 3.4e+264) {
		tmp = x * (sinh(y) / x);
	} else {
		tmp = 1.0 / ((x / y) / (x + (-0.16666666666666666 * pow(x, 3.0))));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.0045d0) then
        tmp = y * (sin(x) / x)
    else if (y <= 3.4d+264) then
        tmp = x * (sinh(y) / x)
    else
        tmp = 1.0d0 / ((x / y) / (x + ((-0.16666666666666666d0) * (x ** 3.0d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0045) {
		tmp = y * (Math.sin(x) / x);
	} else if (y <= 3.4e+264) {
		tmp = x * (Math.sinh(y) / x);
	} else {
		tmp = 1.0 / ((x / y) / (x + (-0.16666666666666666 * Math.pow(x, 3.0))));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0045:
		tmp = y * (math.sin(x) / x)
	elif y <= 3.4e+264:
		tmp = x * (math.sinh(y) / x)
	else:
		tmp = 1.0 / ((x / y) / (x + (-0.16666666666666666 * math.pow(x, 3.0))))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0045)
		tmp = Float64(y * Float64(sin(x) / x));
	elseif (y <= 3.4e+264)
		tmp = Float64(x * Float64(sinh(y) / x));
	else
		tmp = Float64(1.0 / Float64(Float64(x / y) / Float64(x + Float64(-0.16666666666666666 * (x ^ 3.0)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0045)
		tmp = y * (sin(x) / x);
	elseif (y <= 3.4e+264)
		tmp = x * (sinh(y) / x);
	else
		tmp = 1.0 / ((x / y) / (x + (-0.16666666666666666 * (x ^ 3.0))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0045], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.4e+264], N[(x * N[(N[Sinh[y], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(N[(x / y), $MachinePrecision] / N[(x + N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.00449999999999999966

   1. Initial program 86.4%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 45.9%

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

      1. associate-/l* 59.4%

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

   7. Simplified 59.4%

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

   if 0.00449999999999999966 < y < 3.4000000000000001e264

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 76.3%

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

   if 3.4000000000000001e264 < y

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 7.8%

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

      1. associate-/l* 7.8%

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

   7. Simplified 7.8%

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

      1. associate-*r/ 7.8%

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

      2. clear-num 7.8%

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

      3. associate-/r* 64.8%

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

   9. Applied egg-rr 64.8%

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

   10. Taylor expanded in x around 0 84.2%

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

       1. distribute-rgt-in 84.2%

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

       2. *-lft-identity 84.2%

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

       3. associate-*l* 84.2%

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

       4. unpow2 84.2%

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

       5. unpow3 84.2%

          \[\leadsto \frac{1}{\frac{\frac{x}{y}}{x + -0.16666666666666666 \cdot \color{blue}{{x}^{3}}}} \]

   12. Simplified 84.2%

       \[\leadsto \frac{1}{\frac{\frac{x}{y}}{\color{blue}{x + -0.16666666666666666 \cdot {x}^{3}}}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 68.9% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.0014:\\ \;\;\;\;y \cdot \frac{\sin x}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.0014) (* y (/ (sin x) x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.0014) {
		tmp = y * (sin(x) / x);
	} else {
		tmp = sinh(y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.0014d0) then
        tmp = y * (sin(x) / x)
    else
        tmp = sinh(y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.0014) {
		tmp = y * (Math.sin(x) / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.0014:
		tmp = y * (math.sin(x) / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.0014)
		tmp = Float64(y * Float64(sin(x) / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.0014)
		tmp = y * (sin(x) / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.0014], N[(y * N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.00139999999999999999

   1. Initial program 86.4%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in y around 0 45.9%

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

      1. associate-/l* 59.4%

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

   7. Simplified 59.4%

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

   if 0.00139999999999999999 < y

   1. Initial program 100.0%

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

      1. associate-/l* 100.0%

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

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 76.9%

      \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]
   6. Step-by-step derivation

      1. clear-num 76.9%

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

      2. un-div-inv 76.9%

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

   7. Applied egg-rr 76.9%

      \[\leadsto \color{blue}{\frac{x}{\frac{x}{\sinh y}}} \]
   8. Step-by-step derivation

      1. associate-/r/ 76.9%

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

      2. *-inverses 76.9%

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

      3. *-lft-identity 76.9%

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

   9. Simplified 76.9%

      \[\leadsto \color{blue}{\sinh y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 74.2% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (* x (/ (sinh y) x)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 76.4%

   \[\leadsto \color{blue}{x} \cdot \frac{\sinh y}{x} \]
6. Add Preprocessing

Alternative 6: 49.5% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x / (x / y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 33.4%

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

   1. associate-/l* 42.8%

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

7. Simplified 42.8%

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

8. Taylor expanded in y around 0 33.4%

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

   1. associate-*l/ 60.2%

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

   2. *-commutative 60.2%

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

10. Simplified 60.2%

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

11. Taylor expanded in x around 0 51.3%

    \[\leadsto \color{blue}{x} \cdot \frac{y}{x} \]
12. Step-by-step derivation

    1. clear-num 51.8%

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

    2. div-inv 51.5%

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

13. Applied egg-rr 51.5%

    \[\leadsto \color{blue}{\frac{x}{\frac{x}{y}}} \]
14. Add Preprocessing

Alternative 7: 49.7% accurate, 41.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x * (y / x)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.6%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in y around 0 33.4%

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

   1. associate-/l* 42.8%

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

7. Simplified 42.8%

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

8. Taylor expanded in y around 0 33.4%

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

   1. associate-*l/ 60.2%

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

   2. *-commutative 60.2%

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

10. Simplified 60.2%

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

11. Taylor expanded in x around 0 51.3%

    \[\leadsto \color{blue}{x} \cdot \frac{y}{x} \]
12. Add Preprocessing

Alternative 8: 27.0% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 90.6%

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

   1. associate-/l* 99.9%

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

3. Simplified 99.9%

   \[\leadsto \color{blue}{\sin x \cdot \frac{\sinh y}{x}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 76.4%

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

6. Taylor expanded in y around 0 23.7%

   \[\leadsto \color{blue}{y} \]
7. Add Preprocessing

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024091 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))