Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.5% → 99.8%

Time: 9.5s

Alternatives: 10

Speedup: 1.0×

• Could not converge on a ground truth

  1. x = 4.5897090062939974e-20
  2. y = 1359907268947401500.0

• 49.2% 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: 89.5% 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

\[\begin{array}{l} \\ \frac{\frac{\sin x}{x}}{\frac{1}{\sinh y}} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (/ (sin x) x) (/ 1.0 (sinh y))))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return (math.sin(x) / x) / (1.0 / math.sinh(y))

Julia

function code(x, y)
	return Float64(Float64(sin(x) / x) / Float64(1.0 / sinh(y)))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. associate-*r/ 99.9%

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

   2. *-commutative 99.9%

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

   3. div-inv 99.8%

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

   4. associate-*l* 99.9%

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

4. Applied egg-rr 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*l/ 99.9%

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

   3. *-un-lft-identity 99.9%

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

   4. associate-/r/ 99.4%

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

   5. div-inv 99.3%

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

   6. associate-/r* 99.9%

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

6. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\frac{\sin x}{x}}{\frac{1}{\sinh y}}} \]
7. Add Preprocessing

Alternative 2: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2.5e-6) (/ y (/ x (sin x))) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2.5e-6) {
		tmp = y / (x / sin(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 (sinh(y) <= 2.5d-6) then
        tmp = y / (x / sin(x))
    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) <= 2.5e-6) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2.5e-6:
		tmp = y / (x / math.sin(x))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2.5e-6)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2.5e-6)
		tmp = y / (x / sin(x));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2.5e-6], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2.5000000000000002e-6

   1. Initial program 86.0%

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

      1. associate-*r/ 99.8%

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

      2. *-commutative 99.8%

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

      3. div-inv 99.7%

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

      4. associate-*l* 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.9%

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

      3. *-un-lft-identity 99.9%

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

      4. associate-/r/ 99.2%

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

      5. div-inv 99.1%

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

      6. associate-/r* 99.8%

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

   6. Applied egg-rr 99.8%

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

   7. Taylor expanded in y around 0 53.2%

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

      1. associate-*l/ 75.1%

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

      2. associate-/r/ 67.1%

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

   9. Simplified 67.1%

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

   if 2.5000000000000002e-6 < (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 72.6%

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

      1. associate-*r/ 72.6%

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

      2. clear-num 72.6%

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

      3. *-commutative 72.6%

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

   7. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\sinh y \cdot x}}} \]
   8. Step-by-step derivation

      1. *-inverses 72.6%

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

      2. *-commutative 72.6%

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

      3. associate-/r* 72.6%

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

      4. *-inverses 72.6%

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

      5. associate-/r* 72.6%

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

      6. associate-*r/ 72.6%

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

      7. *-rgt-identity 72.6%

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

      8. associate-/r/ 72.6%

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

      9. *-inverses 72.6%

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

      10. *-lft-identity 72.6%

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

   9. Simplified 72.6%

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

Alternative 3: 69.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2.5e-6) (* (/ (sin x) x) y) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2.5e-6) {
		tmp = (sin(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) <= 2.5d-6) then
        tmp = (sin(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) <= 2.5e-6) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2.5e-6:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2.5e-6)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2.5e-6)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2.5e-6], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2.5000000000000002e-6

   1. Initial program 86.0%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in y around 0 53.2%

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

      1. associate-/l* 67.1%

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

   7. Simplified 67.1%

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

   if 2.5000000000000002e-6 < (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 72.6%

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

      1. associate-*r/ 72.6%

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

      2. clear-num 72.6%

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

      3. *-commutative 72.6%

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

   7. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\sinh y \cdot x}}} \]
   8. Step-by-step derivation

      1. *-inverses 72.6%

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

      2. *-commutative 72.6%

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

      3. associate-/r* 72.6%

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

      4. *-inverses 72.6%

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

      5. associate-/r* 72.6%

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

      6. associate-*r/ 72.6%

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

      7. *-rgt-identity 72.6%

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

      8. associate-/r/ 72.6%

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

      9. *-inverses 72.6%

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

      10. *-lft-identity 72.6%

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

   9. Simplified 72.6%

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

4. Final simplification 68.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.8%

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

   1. associate-*r/ 99.9%

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

   2. *-commutative 99.9%

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

   3. div-inv 99.8%

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

   4. associate-*l* 99.9%

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

4. Applied egg-rr 99.9%

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

5. Final simplification 99.9%

   \[\leadsto \sinh y \cdot \left(\sin x \cdot \frac{1}{x}\right) \]
6. Add Preprocessing

Alternative 5: 62.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{x}{\frac{x}{y}}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2.5e-6) (/ x (/ x y)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2.5e-6) {
		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) <= 2.5d-6) 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) <= 2.5e-6) {
		tmp = x / (x / y);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2.5e-6:
		tmp = x / (x / y)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2.5e-6)
		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) <= 2.5e-6)
		tmp = x / (x / y);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2.5e-6], N[(x / N[(x / y), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2.5000000000000002e-6

   1. Initial program 86.0%

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

      1. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 72.8%

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

   6. Taylor expanded in y around 0 55.7%

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

      1. clear-num 57.0%

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

      2. un-div-inv 56.1%

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

   8. Applied egg-rr 56.1%

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

   if 2.5000000000000002e-6 < (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 72.6%

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

      1. associate-*r/ 72.6%

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

      2. clear-num 72.6%

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

      3. *-commutative 72.6%

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

   7. Applied egg-rr 72.6%

      \[\leadsto \color{blue}{\frac{1}{\frac{x}{\sinh y \cdot x}}} \]
   8. Step-by-step derivation

      1. *-inverses 72.6%

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

      2. *-commutative 72.6%

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

      3. associate-/r* 72.6%

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

      4. *-inverses 72.6%

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

      5. associate-/r* 72.6%

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

      6. associate-*r/ 72.6%

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

      7. *-rgt-identity 72.6%

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

      8. associate-/r/ 72.6%

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

      9. *-inverses 72.6%

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

      10. *-lft-identity 72.6%

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

   9. Simplified 72.6%

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

Alternative 6: 99.9% 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 89.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 7: 73.8% 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 89.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 72.8%

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

Alternative 8: 50.0% 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 89.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 72.8%

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

6. Taylor expanded in y around 0 46.8%

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

   1. clear-num 47.8%

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

   2. un-div-inv 47.1%

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

8. Applied egg-rr 47.1%

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

Alternative 9: 50.4% 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 89.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 72.8%

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

6. Taylor expanded in y around 0 46.8%

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

Alternative 10: 27.2% 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 89.8%

   \[\frac{\sin x \cdot \sinh y}{x} \]
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 72.8%

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

6. Taylor expanded in y around 0 24.0%

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

Developer Target 1: 99.9% 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 2024185 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (* (sin x) (/ (sinh y) x)))

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