Linear.Quaternion:$csin from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 6

Speedup: 1.0×

• 49.1% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

   \[\leadsto \cos x \cdot \frac{\sinh y}{y} \]
4. Add Preprocessing

Alternative 2: 55.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+42}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot {x}^{2}\right)}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 6e+42) (cos x) (/ (+ y (* -0.5 (* y (pow x 2.0)))) y)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 6e+42) {
		tmp = cos(x);
	} else {
		tmp = (y + (-0.5 * (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 <= 6d+42) then
        tmp = cos(x)
    else
        tmp = (y + ((-0.5d0) * (y * (x ** 2.0d0)))) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 6e+42) {
		tmp = Math.cos(x);
	} else {
		tmp = (y + (-0.5 * (y * Math.pow(x, 2.0)))) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 6e+42:
		tmp = math.cos(x)
	else:
		tmp = (y + (-0.5 * (y * math.pow(x, 2.0)))) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 6e+42)
		tmp = cos(x);
	else
		tmp = Float64(Float64(y + Float64(-0.5 * Float64(y * (x ^ 2.0)))) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 6e+42)
		tmp = cos(x);
	else
		tmp = (y + (-0.5 * (y * (x ^ 2.0)))) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 6e+42], N[Cos[x], $MachinePrecision], N[(N[(y + N[(-0.5 * N[(y * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.00000000000000058e42

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 70.1%

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

   if 6.00000000000000058e42 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 3.1%

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

   8. Taylor expanded in x around 0 20.0%

      \[\leadsto \frac{\color{blue}{y + -0.5 \cdot \left({x}^{2} \cdot y\right)}}{y} \]
3. Recombined 2 regimes into one program.

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6 \cdot 10^{+42}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;\frac{y + -0.5 \cdot \left(y \cdot {x}^{2}\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{+42}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot {x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.6e+42) (cos x) (+ 1.0 (* -0.5 (pow x 2.0)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.6e+42) {
		tmp = cos(x);
	} else {
		tmp = 1.0 + (-0.5 * 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 <= 7.6d+42) then
        tmp = cos(x)
    else
        tmp = 1.0d0 + ((-0.5d0) * (x ** 2.0d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.6e+42) {
		tmp = Math.cos(x);
	} else {
		tmp = 1.0 + (-0.5 * Math.pow(x, 2.0));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.6e+42:
		tmp = math.cos(x)
	else:
		tmp = 1.0 + (-0.5 * math.pow(x, 2.0))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.6e+42)
		tmp = cos(x);
	else
		tmp = Float64(1.0 + Float64(-0.5 * (x ^ 2.0)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.6e+42)
		tmp = cos(x);
	else
		tmp = 1.0 + (-0.5 * (x ^ 2.0));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.6e+42], N[Cos[x], $MachinePrecision], N[(1.0 + N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.5999999999999997e42

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 70.1%

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

   if 7.5999999999999997e42 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 3.1%

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

   8. Taylor expanded in x around 0 14.9%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.6 \cdot 10^{+42}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;1 + -0.5 \cdot {x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 53.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+54}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {x}^{2}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 7.2e+54) (cos x) (* -0.5 (pow x 2.0))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 7.2e+54) {
		tmp = cos(x);
	} else {
		tmp = -0.5 * 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 <= 7.2d+54) then
        tmp = cos(x)
    else
        tmp = (-0.5d0) * (x ** 2.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 7.2e+54) {
		tmp = Math.cos(x);
	} else {
		tmp = -0.5 * Math.pow(x, 2.0);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 7.2e+54:
		tmp = math.cos(x)
	else:
		tmp = -0.5 * math.pow(x, 2.0)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 7.2e+54)
		tmp = cos(x);
	else
		tmp = Float64(-0.5 * (x ^ 2.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 7.2e+54)
		tmp = cos(x);
	else
		tmp = -0.5 * (x ^ 2.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 7.2e+54], N[Cos[x], $MachinePrecision], N[(-0.5 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 7.2000000000000003e54

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 68.5%

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

   if 7.2000000000000003e54 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-*r/ 100.0%

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

   6. Simplified 100.0%

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

   7. Taylor expanded in y around 0 3.1%

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

   8. Taylor expanded in x around 0 16.1%

      \[\leadsto \color{blue}{1 + -0.5 \cdot {x}^{2}} \]

   9. Taylor expanded in x around inf 15.4%

      \[\leadsto \color{blue}{-0.5 \cdot {x}^{2}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 58.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 7.2 \cdot 10^{+54}:\\ \;\;\;\;\cos x\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot {x}^{2}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 51.4% accurate, 2.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 (cos x))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return math.cos(x)

Julia

function code(x, y)
	return cos(x)
end

MATLAB

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

Wolfram

code[x_, y_] := N[Cos[x], $MachinePrecision]

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in y around 0 55.7%

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

4. Final simplification 55.7%

   \[\leadsto \cos x \]
5. Add Preprocessing

Alternative 6: 28.7% 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%

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

   1. add-log-exp 99.7%

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

   2. *-un-lft-identity 99.7%

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

   3. log-prod 99.7%

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

   4. metadata-eval 99.7%

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

   5. add-log-exp 100.0%

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

4. Applied egg-rr 100.0%

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

   1. +-lft-identity 100.0%

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

   2. associate-*r/ 99.9%

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

6. Simplified 99.9%

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

7. Taylor expanded in y around 0 55.7%

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

8. Taylor expanded in x around 0 35.0%

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

9. Final simplification 35.0%

   \[\leadsto 1 \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024081 
(FPCore (x y)
  :name "Linear.Quaternion:$csin from linear-1.19.1.3"
  :precision binary64
  (* (cos x) (/ (sinh y) y)))