Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 7.2s

Alternatives: 14

Speedup: 1.0×

• 49.9% 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: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \sinh y}{y}\\ \end{array} \end{array} \]

FPCore

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

C

double code(double x, double y) {
	double tmp;
	if ((sinh(y) / y) <= 1.0) {
		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 ((sinh(y) / y) <= 1.0d0) 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 ((Math.sinh(y) / y) <= 1.0) {
		tmp = Math.sin(x);
	} else {
		tmp = (x * Math.sinh(y)) / y;
	}
	return tmp;
}

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if ((sinh(y) / y) <= 1.0)
		tmp = sin(x);
	else
		tmp = (x * sinh(y)) / y;
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 99.9%

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

      1. add-log-exp 96.6%

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

      2. *-un-lft-identity 96.6%

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

      3. log-prod 96.6%

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

      4. metadata-eval 96.6%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 99.9%

         \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

   4. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
   5. Step-by-step derivation

      1. +-lft-identity 99.9%

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

      2. associate-*r/ 99.9%

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

   6. Simplified 99.9%

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

   7. Taylor expanded in x around 0 78.3%

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

Alternative 3: 87.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t\_0 \leq 1:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.0) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.0) {
		tmp = sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = sinh(y) / y
    if (t_0 <= 1.0d0) then
        tmp = sin(x)
    else
        tmp = x * t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = Math.sinh(y) / y;
	double tmp;
	if (t_0 <= 1.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = math.sinh(y) / y
	tmp = 0
	if t_0 <= 1.0:
		tmp = math.sin(x)
	else:
		tmp = x * t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(sinh(y) / y)
	tmp = 0.0
	if (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(x * t_0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = sinh(y) / y;
	tmp = 0.0;
	if (t_0 <= 1.0)
		tmp = sin(x);
	else
		tmp = x * t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[Sinh[y], $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[t$95$0, 1.0], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 100.0%

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

   if 1 < (/.f64 (sinh.f64 y) y)

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0 78.3%

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

Alternative 4: 62.7% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1600:\\ \;\;\;\;\sin x\\ \mathbf{elif}\;y \leq 3.9 \cdot 10^{+88}:\\ \;\;\;\;-0.16666666666666666 \cdot {x}^{3}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot y\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1600.0)
   (sin x)
   (if (<= y 3.9e+88)
     (* -0.16666666666666666 (pow x 3.0))
     (+ x (* (* y y) (* x 0.16666666666666666))))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1600.0) {
		tmp = sin(x);
	} else if (y <= 3.9e+88) {
		tmp = -0.16666666666666666 * pow(x, 3.0);
	} else {
		tmp = x + ((y * y) * (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 <= 1600.0d0) then
        tmp = sin(x)
    else if (y <= 3.9d+88) then
        tmp = (-0.16666666666666666d0) * (x ** 3.0d0)
    else
        tmp = x + ((y * y) * (x * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1600.0) {
		tmp = Math.sin(x);
	} else if (y <= 3.9e+88) {
		tmp = -0.16666666666666666 * Math.pow(x, 3.0);
	} else {
		tmp = x + ((y * y) * (x * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1600.0:
		tmp = math.sin(x)
	elif y <= 3.9e+88:
		tmp = -0.16666666666666666 * math.pow(x, 3.0)
	else:
		tmp = x + ((y * y) * (x * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1600.0)
		tmp = sin(x);
	elseif (y <= 3.9e+88)
		tmp = Float64(-0.16666666666666666 * (x ^ 3.0));
	else
		tmp = Float64(x + Float64(Float64(y * y) * Float64(x * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1600.0)
		tmp = sin(x);
	elseif (y <= 3.9e+88)
		tmp = -0.16666666666666666 * (x ^ 3.0);
	else
		tmp = x + ((y * y) * (x * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1600.0], N[Sin[x], $MachinePrecision], If[LessEqual[y, 3.9e+88], N[(-0.16666666666666666 * N[Power[x, 3.0], $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * y), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < 1600

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.2%

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

   if 1600 < y < 3.9000000000000001e88

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 2.7%

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

   4. Taylor expanded in x around 0 13.5%

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

      1. distribute-rgt-in 13.5%

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

      2. *-lft-identity 13.5%

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

      3. associate-*l* 13.5%

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

      4. pow-plus 13.5%

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

      5. metadata-eval 13.5%

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

   6. Simplified 13.5%

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

   7. Taylor expanded in x around inf 13.1%

      \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3}} \]

   if 3.9000000000000001e88 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 59.7%

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

      1. associate-*r* 59.7%

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

   5. Simplified 59.7%

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

   6. Taylor expanded in x around 0 58.7%

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

      1. *-commutative 58.7%

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

      2. *-commutative 58.7%

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

      3. associate-*r* 58.7%

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

   8. Simplified 58.7%

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

      1. unpow2 58.7%

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

   10. Applied egg-rr 58.7%

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

   11. Taylor expanded in x around 0 58.7%

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

Alternative 5: 61.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.5 \cdot 10^{-10}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot y\right) \cdot \left(x \cdot 0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 1.5e-10) (sin x) (+ x (* (* y y) (* x 0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-10) {
		tmp = sin(x);
	} else {
		tmp = x + ((y * y) * (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 <= 1.5d-10) then
        tmp = sin(x)
    else
        tmp = x + ((y * y) * (x * 0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.5e-10) {
		tmp = Math.sin(x);
	} else {
		tmp = x + ((y * y) * (x * 0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.5e-10:
		tmp = math.sin(x)
	else:
		tmp = x + ((y * y) * (x * 0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.5e-10)
		tmp = sin(x);
	else
		tmp = Float64(x + Float64(Float64(y * y) * Float64(x * 0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.5e-10)
		tmp = sin(x);
	else
		tmp = x + ((y * y) * (x * 0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.5e-10], N[Sin[x], $MachinePrecision], N[(x + N[(N[(y * y), $MachinePrecision] * N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.5e-10

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 65.5%

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

   if 1.5e-10 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 45.4%

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

      1. associate-*r* 45.4%

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

   5. Simplified 45.4%

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

   6. Taylor expanded in x around 0 44.3%

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

      1. *-commutative 44.3%

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

      2. *-commutative 44.3%

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

      3. associate-*r* 44.3%

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

   8. Simplified 44.3%

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

      1. unpow2 44.3%

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

   10. Applied egg-rr 44.3%

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

   11. Taylor expanded in x around 0 43.9%

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

Alternative 6: 48.9% accurate, 22.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 74.7%

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

   1. associate-*r* 74.7%

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

5. Simplified 74.7%

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

6. Taylor expanded in x around 0 65.1%

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

   1. *-commutative 65.1%

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

   2. *-commutative 65.1%

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

   3. associate-*r* 65.1%

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

8. Simplified 65.1%

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

   1. unpow2 65.1%

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

10. Applied egg-rr 65.1%

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

11. Taylor expanded in x around 0 47.3%

    \[\leadsto \color{blue}{x} + \left(y \cdot y\right) \cdot \left(x \cdot 0.16666666666666666\right) \]
12. Add Preprocessing

Alternative 7: 27.6% accurate, 34.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4100000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 4100000.0) x 0.5))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4100000.0) {
		tmp = x;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 4100000.0d0) then
        tmp = x
    else
        tmp = 0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4100000.0) {
		tmp = x;
	} else {
		tmp = 0.5;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4100000.0:
		tmp = x
	else:
		tmp = 0.5
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4100000.0)
		tmp = x;
	else
		tmp = 0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4100000.0)
		tmp = x;
	else
		tmp = 0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4100000.0], x, 0.5]

Derivation

1. Split input into 2 regimes

2. if x < 4.1e6

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 81.4%

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

   4. Taylor expanded in y around 0 31.9%

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

   if 4.1e6 < x

   1. Initial program 100.0%

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

      1. add-log-exp 99.5%

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

      2. *-un-lft-identity 99.5%

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

      3. log-prod 99.5%

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

      4. metadata-eval 99.5%

         \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

      5. add-log-exp 100.0%

         \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
   5. Step-by-step derivation

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 99.7%

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

   6. Simplified 99.7%

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

   8. Applied egg-rr 8.7%

      \[\leadsto \color{blue}{0.5} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 4.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.5)

C

double code(double x, double y) {
	return 0.5;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.5d0
end function

Java

public static double code(double x, double y) {
	return 0.5;
}

Python

def code(x, y):
	return 0.5

Julia

function code(x, y)
	return 0.5
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5;
end

Wolfram

code[x_, y_] := 0.5

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 78.0%

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

   2. *-un-lft-identity 78.0%

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

   3. log-prod 78.0%

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

   4. metadata-eval 78.0%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*r/ 89.5%

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

6. Simplified 89.5%

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

8. Applied egg-rr 4.9%

   \[\leadsto \color{blue}{0.5} \]
9. Add Preprocessing

Alternative 9: 4.5% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.3333333333333333 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.3333333333333333)

C

double code(double x, double y) {
	return 0.3333333333333333;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.3333333333333333d0
end function

Java

public static double code(double x, double y) {
	return 0.3333333333333333;
}

Python

def code(x, y):
	return 0.3333333333333333

Julia

function code(x, y)
	return 0.3333333333333333
end

MATLAB

function tmp = code(x, y)
	tmp = 0.3333333333333333;
end

Wolfram

code[x_, y_] := 0.3333333333333333

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 78.0%

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

   2. *-un-lft-identity 78.0%

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

   3. log-prod 78.0%

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

   4. metadata-eval 78.0%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*r/ 89.5%

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

6. Simplified 89.5%

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

8. Applied egg-rr 4.7%

   \[\leadsto \color{blue}{0.3333333333333333} \]
9. Add Preprocessing

Alternative 10: 4.4% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.25 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.25)

C

double code(double x, double y) {
	return 0.25;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.25d0
end function

Java

public static double code(double x, double y) {
	return 0.25;
}

Python

def code(x, y):
	return 0.25

Julia

function code(x, y)
	return 0.25
end

MATLAB

function tmp = code(x, y)
	tmp = 0.25;
end

Wolfram

code[x_, y_] := 0.25

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 78.0%

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

   2. *-un-lft-identity 78.0%

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

   3. log-prod 78.0%

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

   4. metadata-eval 78.0%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*r/ 89.5%

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

6. Simplified 89.5%

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

8. Applied egg-rr 4.6%

   \[\leadsto \color{blue}{0.25} \]
9. Add Preprocessing

Alternative 11: 4.3% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.16666666666666666 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.16666666666666666)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return 0.16666666666666666

Julia

function code(x, y)
	return 0.16666666666666666
end

MATLAB

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

Wolfram

code[x_, y_] := 0.16666666666666666

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 78.0%

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

   2. *-un-lft-identity 78.0%

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

   3. log-prod 78.0%

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

   4. metadata-eval 78.0%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*r/ 89.5%

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

6. Simplified 89.5%

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

8. Applied egg-rr 4.4%

   \[\leadsto \color{blue}{0.16666666666666666} \]
9. Add Preprocessing

Alternative 12: 4.1% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0.027777777777777776 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.027777777777777776)

C

double code(double x, double y) {
	return 0.027777777777777776;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.027777777777777776d0
end function

Java

public static double code(double x, double y) {
	return 0.027777777777777776;
}

Python

def code(x, y):
	return 0.027777777777777776

Julia

function code(x, y)
	return 0.027777777777777776
end

MATLAB

function tmp = code(x, y)
	tmp = 0.027777777777777776;
end

Wolfram

code[x_, y_] := 0.027777777777777776

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 78.0%

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

   2. *-un-lft-identity 78.0%

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

   3. log-prod 78.0%

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

   4. metadata-eval 78.0%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*r/ 89.5%

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

6. Simplified 89.5%

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

8. Applied egg-rr 4.1%

   \[\leadsto \color{blue}{0.027777777777777776} \]
9. Add Preprocessing

Alternative 13: 4.6% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -1 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

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

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 78.0%

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

   2. *-un-lft-identity 78.0%

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

   3. log-prod 78.0%

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

   4. metadata-eval 78.0%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*r/ 89.5%

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

6. Simplified 89.5%

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

8. Applied egg-rr 5.0%

   \[\leadsto \color{blue}{-1} \]
9. Add Preprocessing

Alternative 14: 4.1% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ -3 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 -3.0)

C

double code(double x, double y) {
	return -3.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -3.0d0
end function

Java

public static double code(double x, double y) {
	return -3.0;
}

Python

def code(x, y):
	return -3.0

Julia

function code(x, y)
	return -3.0
end

MATLAB

function tmp = code(x, y)
	tmp = -3.0;
end

Wolfram

code[x_, y_] := -3.0

Derivation

1. Initial program 100.0%

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

   1. add-log-exp 78.0%

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

   2. *-un-lft-identity 78.0%

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

   3. log-prod 78.0%

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

   4. metadata-eval 78.0%

      \[\leadsto \color{blue}{0} + \log \left(e^{\sin x \cdot \frac{\sinh y}{y}}\right) \]

   5. add-log-exp 100.0%

      \[\leadsto 0 + \color{blue}{\sin x \cdot \frac{\sinh y}{y}} \]

4. Applied egg-rr 100.0%

   \[\leadsto \color{blue}{0 + \sin x \cdot \frac{\sinh y}{y}} \]
5. Step-by-step derivation

   1. +-lft-identity 100.0%

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

   2. associate-*r/ 89.5%

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

6. Simplified 89.5%

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

8. Applied egg-rr 4.3%

   \[\leadsto \color{blue}{-3} \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024130 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))