Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.7s

Alternatives: 7

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

Alternative 2: 86.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.000000000005:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.000000000005) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.000000000005) {
		tmp = sin(y) / y;
	} else {
		tmp = cosh(x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (cosh(x) <= 1.000000000005d0) then
        tmp = sin(y) / y
    else
        tmp = cosh(x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.cosh(x) <= 1.000000000005) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.000000000005:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.000000000005)
		tmp = Float64(sin(y) / y);
	else
		tmp = cosh(x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (cosh(x) <= 1.000000000005)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.000000000005], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.000000000005

   1. Initial program 99.8%

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

   3. Taylor expanded in x around 0 99.8%

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

   if 1.000000000005 < (cosh.f64 x)

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around 0 76.3%

      \[\leadsto \frac{\cosh x}{\color{blue}{1}} \]
   6. Step-by-step derivation

      1. div-inv 76.3%

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

      2. metadata-eval 76.3%

         \[\leadsto \cosh x \cdot \color{blue}{1} \]

   7. Applied egg-rr 76.3%

      \[\leadsto \color{blue}{\cosh x \cdot 1} \]
   8. Step-by-step derivation

      1. *-rgt-identity 76.3%

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

   9. Simplified 76.3%

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

Alternative 3: 62.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 5.3 \cdot 10^{+161}:\\ \;\;\;\;\cosh x\\ \mathbf{else}:\\ \;\;\;\;1 + \left(y \cdot y\right) \cdot -0.16666666666666666\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 5.3e+161) (cosh x) (+ 1.0 (* (* y y) -0.16666666666666666))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 5.3e+161) {
		tmp = cosh(x);
	} else {
		tmp = 1.0 + ((y * y) * -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 <= 5.3d+161) then
        tmp = cosh(x)
    else
        tmp = 1.0d0 + ((y * y) * (-0.16666666666666666d0))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 5.3e+161) {
		tmp = Math.cosh(x);
	} else {
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 5.3e+161:
		tmp = math.cosh(x)
	else:
		tmp = 1.0 + ((y * y) * -0.16666666666666666)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 5.3e+161)
		tmp = cosh(x);
	else
		tmp = Float64(1.0 + Float64(Float64(y * y) * -0.16666666666666666));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 5.3e+161)
		tmp = cosh(x);
	else
		tmp = 1.0 + ((y * y) * -0.16666666666666666);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 5.3e+161], N[Cosh[x], $MachinePrecision], N[(1.0 + N[(N[(y * y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 5.29999999999999955e161

   1. Initial program 99.9%

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

      1. clear-num 99.9%

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

      2. un-div-inv 99.9%

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in y around 0 68.4%

      \[\leadsto \frac{\cosh x}{\color{blue}{1}} \]
   6. Step-by-step derivation

      1. div-inv 68.4%

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

      2. metadata-eval 68.4%

         \[\leadsto \cosh x \cdot \color{blue}{1} \]

   7. Applied egg-rr 68.4%

      \[\leadsto \color{blue}{\cosh x \cdot 1} \]
   8. Step-by-step derivation

      1. *-rgt-identity 68.4%

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

   9. Simplified 68.4%

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

   if 5.29999999999999955e161 < y

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-*l/ 99.8%

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

      3. associate-/l* 99.7%

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

   3. Simplified 99.7%

      \[\leadsto \color{blue}{\sin y \cdot \frac{\cosh x}{y}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 54.3%

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

   6. Taylor expanded in y around 0 27.7%

      \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}} \]
   7. Step-by-step derivation

      1. *-commutative 27.7%

         \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

   8. Simplified 27.7%

      \[\leadsto \color{blue}{1 + {y}^{2} \cdot -0.16666666666666666} \]
   9. Step-by-step derivation

      1. unpow2 27.7%

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

   10. Applied egg-rr 27.7%

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

Alternative 4: 31.9% accurate, 29.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*l/ 99.9%

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

   3. associate-/l* 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\sin y \cdot \frac{\cosh x}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 43.6%

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

6. Taylor expanded in y around 0 27.3%

   \[\leadsto \color{blue}{1 + -0.16666666666666666 \cdot {y}^{2}} \]
7. Step-by-step derivation

   1. *-commutative 27.3%

      \[\leadsto 1 + \color{blue}{{y}^{2} \cdot -0.16666666666666666} \]

8. Simplified 27.3%

   \[\leadsto \color{blue}{1 + {y}^{2} \cdot -0.16666666666666666} \]
9. Step-by-step derivation

   1. unpow2 27.3%

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

10. Applied egg-rr 27.3%

    \[\leadsto 1 + \color{blue}{\left(y \cdot y\right)} \cdot -0.16666666666666666 \]
11. Add Preprocessing

Alternative 5: 26.3% accurate, 41.0× speedup

Math

\[\begin{array}{l} \\ y \cdot \frac{1}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* y (/ 1.0 y)))

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y * (1.0 / y)

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*l/ 99.9%

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

   3. associate-/l* 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\sin y \cdot \frac{\cosh x}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 43.6%

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

6. Taylor expanded in y around 0 21.0%

   \[\leadsto \color{blue}{y} \cdot \frac{1}{y} \]
7. Add Preprocessing

Alternative 6: 26.3% 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 99.9%

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

   1. clear-num 99.9%

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

   2. un-div-inv 99.9%

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

4. Applied egg-rr 99.9%

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

5. Taylor expanded in y around 0 63.6%

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

6. Taylor expanded in x around 0 20.7%

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

Alternative 7: 2.9% accurate, 205.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 99.9%

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

   1. *-commutative 99.9%

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

   2. associate-*l/ 99.9%

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

   3. associate-/l* 99.4%

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

3. Simplified 99.4%

   \[\leadsto \color{blue}{\sin y \cdot \frac{\cosh x}{y}} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 43.6%

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

   1. un-div-inv 43.3%

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

   2. expm1-log1p-u 43.3%

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

   3. expm1-undefine 25.4%

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

   4. div-sub 25.4%

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

   5. log1p-undefine 25.4%

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

   6. rem-exp-log 25.4%

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

   7. +-commutative 25.4%

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

7. Applied egg-rr 25.4%

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

8. Taylor expanded in y around 0 2.8%

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

9. Taylor expanded in y around 0 2.8%

   \[\leadsto \color{blue}{0} \]
10. Add Preprocessing

Developer Target 1: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024181 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :alt
  (! :herbie-platform default (/ (* (cosh x) (sin y)) y))

  (* (cosh x) (/ (sin y) y)))