Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.9s

Alternatives: 10

Speedup: 1.0×

• 50.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

2. Final simplification 100.0%

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

Alternative 2: 68.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.082:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+171}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(\sin x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.082)
   (sin x)
   (if (<= y 8.5e+171)
     (* x (/ (sinh y) y))
     (* 0.16666666666666666 (* y (* (sin x) y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.082) {
		tmp = sin(x);
	} else if (y <= 8.5e+171) {
		tmp = x * (sinh(y) / y);
	} else {
		tmp = 0.16666666666666666 * (y * (sin(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 (y <= 0.082d0) then
        tmp = sin(x)
    else if (y <= 8.5d+171) then
        tmp = x * (sinh(y) / y)
    else
        tmp = 0.16666666666666666d0 * (y * (sin(x) * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.082) {
		tmp = Math.sin(x);
	} else if (y <= 8.5e+171) {
		tmp = x * (Math.sinh(y) / y);
	} else {
		tmp = 0.16666666666666666 * (y * (Math.sin(x) * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.082:
		tmp = math.sin(x)
	elif y <= 8.5e+171:
		tmp = x * (math.sinh(y) / y)
	else:
		tmp = 0.16666666666666666 * (y * (math.sin(x) * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.082)
		tmp = sin(x);
	elseif (y <= 8.5e+171)
		tmp = Float64(x * Float64(sinh(y) / y));
	else
		tmp = Float64(0.16666666666666666 * Float64(y * Float64(sin(x) * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.082)
		tmp = sin(x);
	elseif (y <= 8.5e+171)
		tmp = x * (sinh(y) / y);
	else
		tmp = 0.16666666666666666 * (y * (sin(x) * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.082], N[Sin[x], $MachinePrecision], If[LessEqual[y, 8.5e+171], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(y * N[(N[Sin[x], $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.0820000000000000034

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 65.7%

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

   if 0.0820000000000000034 < y < 8.4999999999999995e171

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 91.7%

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

   3. Simplified 91.7%

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

   4. Taylor expanded in x around 0 72.2%

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

      1. associate-/r/ 80.6%

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

   6. Applied egg-rr 80.6%

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

   if 8.4999999999999995e171 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 100.0%

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

      1. unpow2 100.0%

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

   4. Simplified 100.0%

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

   5. Taylor expanded in y around inf 100.0%

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

      1. unpow2 100.0%

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

      2. associate-*l* 82.6%

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

   7. Simplified 82.6%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(y \cdot \left(y \cdot \sin x\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.082:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{+171}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(\sin x \cdot y\right)\right)\\ \end{array} \]

Alternative 3: 71.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.14:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.14)
   (sin x)
   (if (<= y 1e+154)
     (* x (/ (sinh y) y))
     (* 0.16666666666666666 (* (sin x) (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.14) {
		tmp = sin(x);
	} else if (y <= 1e+154) {
		tmp = x * (sinh(y) / y);
	} else {
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.14d0) then
        tmp = sin(x)
    else if (y <= 1d+154) then
        tmp = x * (sinh(y) / y)
    else
        tmp = 0.16666666666666666d0 * (sin(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.14) {
		tmp = Math.sin(x);
	} else if (y <= 1e+154) {
		tmp = x * (Math.sinh(y) / y);
	} else {
		tmp = 0.16666666666666666 * (Math.sin(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.14:
		tmp = math.sin(x)
	elif y <= 1e+154:
		tmp = x * (math.sinh(y) / y)
	else:
		tmp = 0.16666666666666666 * (math.sin(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.14)
		tmp = sin(x);
	elseif (y <= 1e+154)
		tmp = Float64(x * Float64(sinh(y) / y));
	else
		tmp = Float64(0.16666666666666666 * Float64(sin(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.14)
		tmp = sin(x);
	elseif (y <= 1e+154)
		tmp = x * (sinh(y) / y);
	else
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.14], N[Sin[x], $MachinePrecision], If[LessEqual[y, 1e+154], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.14000000000000001

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 65.7%

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

   if 0.14000000000000001 < y < 1.00000000000000004e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around 0 73.3%

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

      1. associate-/r/ 83.3%

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

   6. Applied egg-rr 83.3%

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

   if 1.00000000000000004e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.8%

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

      1. unpow2 96.8%

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

   4. Simplified 96.8%

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

   5. Taylor expanded in y around inf 96.8%

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

      1. unpow2 96.8%

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

      2. *-commutative 96.8%

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

   7. Simplified 96.8%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.14:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 4: 84.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.088:\\ \;\;\;\;\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.088)
   (* (sin x) (+ 1.0 (* 0.16666666666666666 (* y y))))
   (if (<= y 1e+154)
     (* x (/ (sinh y) y))
     (* 0.16666666666666666 (* (sin x) (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.088) {
		tmp = sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 1e+154) {
		tmp = x * (sinh(y) / y);
	} else {
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.088d0) then
        tmp = sin(x) * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    else if (y <= 1d+154) then
        tmp = x * (sinh(y) / y)
    else
        tmp = 0.16666666666666666d0 * (sin(x) * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.088) {
		tmp = Math.sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	} else if (y <= 1e+154) {
		tmp = x * (Math.sinh(y) / y);
	} else {
		tmp = 0.16666666666666666 * (Math.sin(x) * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.088:
		tmp = math.sin(x) * (1.0 + (0.16666666666666666 * (y * y)))
	elif y <= 1e+154:
		tmp = x * (math.sinh(y) / y)
	else:
		tmp = 0.16666666666666666 * (math.sin(x) * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.088)
		tmp = Float64(sin(x) * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	elseif (y <= 1e+154)
		tmp = Float64(x * Float64(sinh(y) / y));
	else
		tmp = Float64(0.16666666666666666 * Float64(sin(x) * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.088)
		tmp = sin(x) * (1.0 + (0.16666666666666666 * (y * y)));
	elseif (y <= 1e+154)
		tmp = x * (sinh(y) / y);
	else
		tmp = 0.16666666666666666 * (sin(x) * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.088], N[(N[Sin[x], $MachinePrecision] * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1e+154], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(0.16666666666666666 * N[(N[Sin[x], $MachinePrecision] * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 0.087999999999999995

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 82.5%

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

      1. unpow2 82.5%

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

   4. Simplified 82.5%

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

   if 0.087999999999999995 < y < 1.00000000000000004e154

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 90.0%

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

   3. Simplified 90.0%

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

   4. Taylor expanded in x around 0 73.3%

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

      1. associate-/r/ 83.3%

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

   6. Applied egg-rr 83.3%

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

   if 1.00000000000000004e154 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 96.8%

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

      1. unpow2 96.8%

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

   4. Simplified 96.8%

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

   5. Taylor expanded in y around inf 96.8%

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

      1. unpow2 96.8%

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

      2. *-commutative 96.8%

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

   7. Simplified 96.8%

      \[\leadsto \color{blue}{0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.088:\\ \;\;\;\;\sin x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \mathbf{elif}\;y \leq 10^{+154}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(\sin x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 5: 68.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.082:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.082) (sin x) (* x (/ (sinh y) y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.082) {
		tmp = sin(x);
	} else {
		tmp = x * (sinh(y) / y);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.082d0) then
        tmp = sin(x)
    else
        tmp = x * (sinh(y) / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.082) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (Math.sinh(y) / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.082:
		tmp = math.sin(x)
	else:
		tmp = x * (math.sinh(y) / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.082)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(sinh(y) / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.082)
		tmp = sin(x);
	else
		tmp = x * (sinh(y) / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.082], N[Sin[x], $MachinePrecision], N[(x * N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.0820000000000000034

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 65.7%

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

   if 0.0820000000000000034 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in x around 0 57.9%

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

      1. associate-/r/ 73.7%

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

   6. Applied egg-rr 73.7%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.082:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \]

Alternative 6: 61.2% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 320:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 320.0) (sin x) (* x (+ 1.0 (* 0.16666666666666666 (* y y))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 320.0) {
		tmp = sin(x);
	} else {
		tmp = x * (1.0 + (0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 320.0d0) then
        tmp = sin(x)
    else
        tmp = x * (1.0d0 + (0.16666666666666666d0 * (y * y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 320.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (1.0 + (0.16666666666666666 * (y * y)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 320.0:
		tmp = math.sin(x)
	else:
		tmp = x * (1.0 + (0.16666666666666666 * (y * y)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 320.0)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 320.0)
		tmp = sin(x);
	else
		tmp = x * (1.0 + (0.16666666666666666 * (y * y)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 320.0], N[Sin[x], $MachinePrecision], N[(x * N[(1.0 + N[(0.16666666666666666 * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 320

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 65.7%

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

   if 320 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 84.2%

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

   3. Simplified 84.2%

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

   4. Taylor expanded in x around 0 57.9%

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

      1. associate-/r/ 73.7%

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

   6. Applied egg-rr 73.7%

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

   7. Taylor expanded in y around 0 45.3%

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

      1. unpow2 48.4%

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

   9. Simplified 45.3%

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

4. Final simplification 61.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 320:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 7: 33.8% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.75 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(x \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.75e-20) x (* 0.16666666666666666 (* y (* x y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.75e-20) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (y * (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 (y <= 3.75d-20) then
        tmp = x
    else
        tmp = 0.16666666666666666d0 * (y * (x * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.75e-20) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (y * (x * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.75e-20:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (y * (x * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.75e-20)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(y * Float64(x * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.75e-20)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (y * (x * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.75e-20], x, N[(0.16666666666666666 * N[(y * N[(x * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.74999999999999991e-20

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in x around 0 56.9%

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

   5. Taylor expanded in y around 0 37.1%

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

   if 3.74999999999999991e-20 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 50.2%

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

      1. unpow2 50.2%

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

   4. Simplified 50.2%

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

   5. Taylor expanded in y around inf 47.1%

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

      1. unpow2 47.1%

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

      2. associate-*l* 37.8%

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

   7. Simplified 37.8%

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

   8. Taylor expanded in x around 0 43.9%

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

      1. unpow2 43.9%

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

      2. associate-*l* 34.6%

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

   10. Simplified 34.6%

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

4. Final simplification 36.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.75 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(y \cdot \left(x \cdot y\right)\right)\\ \end{array} \]

Alternative 8: 36.9% accurate, 22.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3.75 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3.75e-20) x (* 0.16666666666666666 (* x (* y y)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3.75e-20) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (y * y));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 3.75d-20) then
        tmp = x
    else
        tmp = 0.16666666666666666d0 * (x * (y * y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3.75e-20) {
		tmp = x;
	} else {
		tmp = 0.16666666666666666 * (x * (y * y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3.75e-20:
		tmp = x
	else:
		tmp = 0.16666666666666666 * (x * (y * y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3.75e-20)
		tmp = x;
	else
		tmp = Float64(0.16666666666666666 * Float64(x * Float64(y * y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3.75e-20)
		tmp = x;
	else
		tmp = 0.16666666666666666 * (x * (y * y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3.75e-20], x, N[(0.16666666666666666 * N[(x * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.74999999999999991e-20

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-/r/ 93.8%

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

   3. Simplified 93.8%

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

   4. Taylor expanded in x around 0 56.9%

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

   5. Taylor expanded in y around 0 37.1%

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

   if 3.74999999999999991e-20 < y

   1. Initial program 100.0%

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

   2. Taylor expanded in y around 0 50.2%

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

      1. unpow2 50.2%

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

   4. Simplified 50.2%

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

   5. Taylor expanded in y around inf 47.1%

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

      1. unpow2 47.1%

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

      2. associate-*l* 37.8%

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

   7. Simplified 37.8%

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

   8. Taylor expanded in x around 0 43.9%

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

      1. unpow2 43.9%

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

   10. Simplified 43.9%

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

4. Final simplification 38.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3.75 \cdot 10^{-20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.16666666666666666 \cdot \left(x \cdot \left(y \cdot y\right)\right)\\ \end{array} \]

Alternative 9: 48.5% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (* x (+ 1.0 (* 0.16666666666666666 (* y y)))))

C

double code(double x, double y) {
	return x * (1.0 + (0.16666666666666666 * (y * y)));
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = x * (1.0d0 + (0.16666666666666666d0 * (y * y)))
end function

Java

public static double code(double x, double y) {
	return x * (1.0 + (0.16666666666666666 * (y * y)));
}

Python

def code(x, y):
	return x * (1.0 + (0.16666666666666666 * (y * y)))

Julia

function code(x, y)
	return Float64(x * Float64(1.0 + Float64(0.16666666666666666 * Float64(y * y))))
end

MATLAB

function tmp = code(x, y)
	tmp = x * (1.0 + (0.16666666666666666 * (y * y)));
end

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/r/ 91.7%

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

3. Simplified 91.7%

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

4. Taylor expanded in x around 0 56.7%

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

   1. associate-/r/ 65.0%

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

6. Applied egg-rr 65.0%

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

7. Taylor expanded in y around 0 49.4%

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

   1. unpow2 74.9%

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

9. Simplified 49.4%

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

10. Final simplification 49.4%

    \[\leadsto x \cdot \left(1 + 0.16666666666666666 \cdot \left(y \cdot y\right)\right) \]

Alternative 10: 26.4% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 x)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return x

Julia

function code(x, y)
	return x
end

MATLAB

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

Wolfram

code[x_, y_] := x

Derivation

1. Initial program 100.0%

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

   1. *-commutative 100.0%

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

   2. associate-/r/ 91.7%

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

3. Simplified 91.7%

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

4. Taylor expanded in x around 0 56.7%

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

5. Taylor expanded in y around 0 29.2%

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

6. Final simplification 29.2%

   \[\leadsto x \]

Reproduce

herbie shell --seed 2023275 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))