Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 88.7% → 99.9%

Time: 6.5s

Alternatives: 9

Speedup: 1.0×

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

Specification

Math

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

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: 88.7% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 89.4%

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

   1. associate-*l/ 99.9%

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

3. Simplified 99.9%

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

4. Final simplification 99.9%

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

Alternative 2: 75.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-8}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 1e-8) (* (sin x) (/ y x)) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 1e-8) {
		tmp = sin(x) * (y / 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) <= 1d-8) then
        tmp = sin(x) * (y / 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) <= 1e-8) {
		tmp = Math.sin(x) * (y / x);
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 1e-8:
		tmp = math.sin(x) * (y / x)
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 1e-8)
		tmp = Float64(sin(x) * Float64(y / x));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 1e-8)
		tmp = sin(x) * (y / x);
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-8], N[(N[Sin[x], $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 1e-8

   1. Initial program 85.3%

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

      1. *-commutative 85.3%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 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. Taylor expanded in y around 0 53.7%

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

      1. associate-/l* 68.3%

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

      2. associate-/r/ 78.4%

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

   6. Simplified 78.4%

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

   if 1e-8 < (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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 77.8%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 78.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-8}:\\ \;\;\;\;\sin x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 3: 69.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 1e-8) (* (/ (sin x) x) y) (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 1e-8) {
		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) <= 1d-8) 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) <= 1e-8) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 1e-8:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 1e-8)
		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) <= 1e-8)
		tmp = (sin(x) / x) * y;
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-8], 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) < 1e-8

   1. Initial program 85.3%

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

      1. associate-*l/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in y around 0 68.3%

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

   if 1e-8 < (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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 77.8%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-8}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 4: 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 89.4%

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

   1. *-commutative 89.4%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 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. Final simplification 99.9%

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

Alternative 5: 57.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 2e-17)
   (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x))))
   (sinh y)))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 2e-17) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / 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) <= 2d-17) then
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / 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) <= 2e-17) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	} else {
		tmp = Math.sinh(y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 2e-17:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	else:
		tmp = math.sinh(y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 2e-17)
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	else
		tmp = sinh(y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 2e-17)
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	else
		tmp = sinh(y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 2e-17], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Sinh[y], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 2.00000000000000014e-17

   1. Initial program 85.2%

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

      1. *-commutative 85.2%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 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. Taylor expanded in y around 0 53.4%

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

      1. associate-/l* 68.1%

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

      2. associate-/r/ 78.2%

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

   6. Simplified 78.2%

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

      1. associate-/r/ 68.1%

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

      2. div-inv 68.0%

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

      3. associate-/r* 78.2%

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

   8. Applied egg-rr 78.2%

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

   9. Taylor expanded in x around 0 58.5%

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

   10. Taylor expanded in y around 0 48.3%

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

   if 2.00000000000000014e-17 < (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%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 78.1%

      \[\leadsto \color{blue}{1} \cdot \sinh y \]
3. Recombined 2 regimes into one program.

4. Final simplification 56.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\sinh y \leq 2 \cdot 10^{-17}:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\sinh y\\ \end{array} \]

Alternative 6: 48.5% accurate, 15.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 215:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 215.0)
   (/ y (* x (+ (* x 0.16666666666666666) (/ 1.0 x))))
   (/ (/ y x) (* x 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 215.0) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	} else {
		tmp = (y / x) / (x * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 215.0d0) then
        tmp = y / (x * ((x * 0.16666666666666666d0) + (1.0d0 / x)))
    else
        tmp = (y / x) / (x * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 215.0) {
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	} else {
		tmp = (y / x) / (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 215.0:
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)))
	else:
		tmp = (y / x) / (x * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 215.0)
		tmp = Float64(y / Float64(x * Float64(Float64(x * 0.16666666666666666) + Float64(1.0 / x))));
	else
		tmp = Float64(Float64(y / x) / Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 215.0)
		tmp = y / (x * ((x * 0.16666666666666666) + (1.0 / x)));
	else
		tmp = (y / x) / (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 215.0], N[(y / N[(x * N[(N[(x * 0.16666666666666666), $MachinePrecision] + N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 215

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 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. Taylor expanded in y around 0 53.5%

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

      1. associate-/l* 68.0%

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

      2. associate-/r/ 78.0%

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

   6. Simplified 78.0%

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

      1. associate-/r/ 68.0%

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

      2. div-inv 67.9%

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

      3. associate-/r* 77.9%

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

   8. Applied egg-rr 77.9%

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

   9. Taylor expanded in x around 0 58.4%

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

   10. Taylor expanded in y around 0 48.4%

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

   if 215 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-commutative 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. Taylor expanded in y around 0 4.2%

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

      1. associate-/l* 4.2%

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

      2. associate-/r/ 25.4%

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

   6. Simplified 25.4%

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

      1. associate-/r/ 4.2%

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

      2. div-inv 4.2%

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

      3. associate-/r* 25.4%

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

   8. Applied egg-rr 25.4%

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

   9. Taylor expanded in x around 0 24.6%

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

   10. Taylor expanded in x around inf 38.4%

       \[\leadsto \frac{\frac{y}{x}}{\color{blue}{0.16666666666666666 \cdot x}} \]
   11. Step-by-step derivation

       1. *-commutative 38.4%

          \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x \cdot 0.16666666666666666}} \]

   12. Simplified 38.4%

       \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x \cdot 0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 45.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 215:\\ \;\;\;\;\frac{y}{x \cdot \left(x \cdot 0.16666666666666666 + \frac{1}{x}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666}\\ \end{array} \]

Alternative 7: 53.3% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 260:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 260.0) (* x (/ y x)) (/ (/ y x) (* x 0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 260.0) {
		tmp = x * (y / x);
	} else {
		tmp = (y / x) / (x * 0.16666666666666666);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 260.0d0) then
        tmp = x * (y / x)
    else
        tmp = (y / x) / (x * 0.16666666666666666d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 260.0) {
		tmp = x * (y / x);
	} else {
		tmp = (y / x) / (x * 0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 260.0:
		tmp = x * (y / x)
	else:
		tmp = (y / x) / (x * 0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 260.0)
		tmp = Float64(x * Float64(y / x));
	else
		tmp = Float64(Float64(y / x) / Float64(x * 0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 260.0)
		tmp = x * (y / x);
	else
		tmp = (y / x) / (x * 0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 260.0], N[(x * N[(y / x), $MachinePrecision]), $MachinePrecision], N[(N[(y / x), $MachinePrecision] / N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 260

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 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. Taylor expanded in y around 0 53.5%

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

      1. associate-/l* 68.0%

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

      2. associate-/r/ 78.0%

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

   6. Simplified 78.0%

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

      1. associate-/r/ 68.0%

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

      2. clear-num 66.6%

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

   8. Applied egg-rr 66.6%

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

   9. Taylor expanded in x around 0 35.5%

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

       1. clear-num 35.6%

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

       2. lft-mult-inverse 35.6%

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

       3. associate-/l/ 55.7%

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

       4. associate-/r/ 55.7%

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

       5. /-rgt-identity 55.7%

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

   11. Applied egg-rr 55.7%

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

   if 260 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

      3. *-commutative 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. Taylor expanded in y around 0 4.2%

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

      1. associate-/l* 4.2%

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

      2. associate-/r/ 25.4%

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

   6. Simplified 25.4%

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

      1. associate-/r/ 4.2%

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

      2. div-inv 4.2%

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

      3. associate-/r* 25.4%

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

   8. Applied egg-rr 25.4%

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

   9. Taylor expanded in x around 0 24.6%

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

   10. Taylor expanded in x around inf 38.4%

       \[\leadsto \frac{\frac{y}{x}}{\color{blue}{0.16666666666666666 \cdot x}} \]
   11. Step-by-step derivation

       1. *-commutative 38.4%

          \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x \cdot 0.16666666666666666}} \]

   12. Simplified 38.4%

       \[\leadsto \frac{\frac{y}{x}}{\color{blue}{x \cdot 0.16666666666666666}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 50.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 260:\\ \;\;\;\;x \cdot \frac{y}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{y}{x}}{x \cdot 0.16666666666666666}\\ \end{array} \]

Alternative 8: 50.3% 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 89.4%

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

   1. *-commutative 89.4%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 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. Taylor expanded in y around 0 39.8%

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

   1. associate-/l* 50.3%

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

   2. associate-/r/ 63.4%

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

6. Simplified 63.4%

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

   1. associate-/r/ 50.3%

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

   2. clear-num 49.3%

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

8. Applied egg-rr 49.3%

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

9. Taylor expanded in x around 0 26.8%

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

    1. clear-num 26.8%

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

    2. lft-mult-inverse 26.8%

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

    3. associate-/l/ 47.2%

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

    4. associate-/r/ 47.2%

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

    5. /-rgt-identity 47.2%

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

11. Applied egg-rr 47.2%

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

12. Final simplification 47.2%

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

Alternative 9: 28.6% 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 89.4%

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

   1. *-commutative 89.4%

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

   2. associate-*l/ 99.9%

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

   3. *-commutative 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. Taylor expanded in y around 0 39.8%

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

   1. associate-/l* 50.3%

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

   2. associate-/r/ 63.4%

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

6. Simplified 63.4%

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

7. Taylor expanded in x around 0 26.8%

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

8. Final simplification 26.8%

   \[\leadsto y \]

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 2023333 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

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

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