Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 5.7s

Alternatives: 7

Speedup: 1.0×

• 50.3% 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. Add Preprocessing
3. Add Preprocessing

Alternative 2: 55.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 580:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 580.0)
   (sin x)
   (/ (* x (+ y (* -0.16666666666666666 (* y (pow x 2.0))))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 580.0) {
		tmp = sin(x);
	} else {
		tmp = (x * (y + (-0.16666666666666666 * (y * pow(x, 2.0))))) / 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 <= 580.0d0) then
        tmp = sin(x)
    else
        tmp = (x * (y + ((-0.16666666666666666d0) * (y * (x ** 2.0d0))))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 580.0) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * (y + (-0.16666666666666666 * (y * Math.pow(x, 2.0))))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 580.0:
		tmp = math.sin(x)
	else:
		tmp = (x * (y + (-0.16666666666666666 * (y * math.pow(x, 2.0))))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 580.0)
		tmp = sin(x);
	else
		tmp = Float64(Float64(x * Float64(y + Float64(-0.16666666666666666 * Float64(y * (x ^ 2.0))))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 580.0)
		tmp = sin(x);
	else
		tmp = (x * (y + (-0.16666666666666666 * (y * (x ^ 2.0))))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 580.0], N[Sin[x], $MachinePrecision], N[(N[(x * N[(y + N[(-0.16666666666666666 * N[(y * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 580

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 68.6%

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

   if 580 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.6%

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

      1. *-commutative 2.6%

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

   7. Simplified 2.6%

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

   8. Taylor expanded in x around 0 21.1%

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

4. Final simplification 57.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 580:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + -0.16666666666666666 \cdot \left(y \cdot {x}^{2}\right)\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 54.0% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 580:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot {x}^{2}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 580.0) (sin x) (* x (+ 1.0 (* -0.16666666666666666 (pow x 2.0))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 580.0) {
		tmp = sin(x);
	} else {
		tmp = x * (1.0 + (-0.16666666666666666 * pow(x, 2.0)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 580.0d0) then
        tmp = sin(x)
    else
        tmp = x * (1.0d0 + ((-0.16666666666666666d0) * (x ** 2.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 580.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (1.0 + (-0.16666666666666666 * Math.pow(x, 2.0)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 580.0:
		tmp = math.sin(x)
	else:
		tmp = x * (1.0 + (-0.16666666666666666 * math.pow(x, 2.0)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 580.0)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * (x ^ 2.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 580.0)
		tmp = sin(x);
	else
		tmp = x * (1.0 + (-0.16666666666666666 * (x ^ 2.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 580.0], N[Sin[x], $MachinePrecision], N[(x * N[(1.0 + N[(-0.16666666666666666 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 580

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 68.6%

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

   if 580 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.6%

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

      1. *-commutative 2.6%

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

   7. Simplified 2.6%

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

   8. Taylor expanded in x around 0 18.2%

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

Alternative 4: 53.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.52 \cdot 10^{+63}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.52e+63) (sin x) (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.52e+63) {
		tmp = sin(x);
	} else {
		tmp = (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 <= 1.52d+63) then
        tmp = sin(x)
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.52e+63) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.52e+63:
		tmp = math.sin(x)
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.52e+63)
		tmp = sin(x);
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.52e+63)
		tmp = sin(x);
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.52e+63], N[Sin[x], $MachinePrecision], N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.51999999999999993e63

   1. Initial program 100.0%

      \[\sin x \cdot \frac{\sinh y}{y} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 63.9%

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

   if 1.51999999999999993e63 < y

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 2.6%

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

      1. *-commutative 2.6%

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

   7. Simplified 2.6%

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

   8. Taylor expanded in x around 0 14.8%

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

Alternative 5: 28.9% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 10^{+102}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x \cdot y\right) + -1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x 1e+102) x (/ (+ (+ 1.0 (* x y)) -1.0) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 1e+102) {
		tmp = x;
	} else {
		tmp = ((1.0 + (x * y)) + -1.0) / y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 1d+102) then
        tmp = x
    else
        tmp = ((1.0d0 + (x * y)) + (-1.0d0)) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 1e+102) {
		tmp = x;
	} else {
		tmp = ((1.0 + (x * y)) + -1.0) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 1e+102:
		tmp = x
	else:
		tmp = ((1.0 + (x * y)) + -1.0) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 1e+102)
		tmp = x;
	else
		tmp = Float64(Float64(Float64(1.0 + Float64(x * y)) + -1.0) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 1e+102)
		tmp = x;
	else
		tmp = ((1.0 + (x * y)) + -1.0) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 1e+102], x, N[(N[(N[(1.0 + N[(x * y), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 9.99999999999999977e101

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 83.6%

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

   4. Applied egg-rr 83.6%

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

   5. Taylor expanded in y around 0 35.5%

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

      1. *-commutative 35.5%

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

   7. Simplified 35.5%

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

   8. Taylor expanded in x around 0 34.2%

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

   if 9.99999999999999977e101 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 56.5%

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

      1. *-commutative 56.5%

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

   7. Simplified 56.5%

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

      1. expm1-log1p-u 55.9%

         \[\leadsto \frac{\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\sin x \cdot y\right)\right)}}{y} \]

      2. expm1-undefine 4.9%

         \[\leadsto \frac{\color{blue}{e^{\mathsf{log1p}\left(\sin x \cdot y\right)} - 1}}{y} \]

      3. log1p-undefine 4.9%

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

      4. rem-exp-log 5.6%

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

   9. Applied egg-rr 5.6%

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

   10. Taylor expanded in x around 0 12.8%

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

4. Final simplification 29.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 10^{+102}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(1 + x \cdot y\right) + -1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 28.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 8.5 \cdot 10^{+101}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 8.5e+101) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 8.5e+101) {
		tmp = x;
	} else {
		tmp = (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 (x <= 8.5d+101) then
        tmp = x
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 8.5e+101) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 8.5e+101:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 8.5e+101)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 8.5e+101)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 8.5e+101], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 8.5000000000000001e101

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 83.5%

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

   4. Applied egg-rr 83.5%

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

   5. Taylor expanded in y around 0 35.1%

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

      1. *-commutative 35.1%

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

   7. Simplified 35.1%

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

   8. Taylor expanded in x around 0 34.4%

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

   if 8.5000000000000001e101 < x

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

      2. associate-*l/ 99.8%

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

   4. Applied egg-rr 99.8%

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

   5. Taylor expanded in y around 0 57.3%

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

      1. *-commutative 57.3%

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

   7. Simplified 57.3%

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

   8. Taylor expanded in x around 0 12.6%

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

Alternative 7: 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. Add Preprocessing
3. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. associate-*l/ 87.1%

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

4. Applied egg-rr 87.1%

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

5. Taylor expanded in y around 0 40.0%

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

   1. *-commutative 40.0%

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

7. Simplified 40.0%

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

8. Taylor expanded in x around 0 27.4%

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

Reproduce

herbie shell --seed 2024084 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))