Linear.Quaternion:$csinh from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.8%

Time: 8.2s

Alternatives: 6

Speedup: 1.0×

• 49.7% 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: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/r/ 99.6%

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

3. Simplified 99.6%

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

4. Final simplification 99.6%

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

Alternative 2: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.00002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (cosh x) 1.00002) (/ (sin y) y) (cosh x)))

C

double code(double x, double y) {
	double tmp;
	if (cosh(x) <= 1.00002) {
		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) <= 1.00002d0) 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) <= 1.00002) {
		tmp = Math.sin(y) / y;
	} else {
		tmp = Math.cosh(x);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.cosh(x) <= 1.00002:
		tmp = math.sin(y) / y
	else:
		tmp = math.cosh(x)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (cosh(x) <= 1.00002)
		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) <= 1.00002)
		tmp = sin(y) / y;
	else
		tmp = cosh(x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Cosh[x], $MachinePrecision], 1.00002], N[(N[Sin[y], $MachinePrecision] / y), $MachinePrecision], N[Cosh[x], $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (cosh.f64 x) < 1.00001999999999991

   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-/r/ 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around 0 98.5%

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

   if 1.00001999999999991 < (cosh.f64 x)

   1. Initial program 99.3%

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

      1. *-commutative 99.3%

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

      2. associate-/r/ 99.3%

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

   3. Simplified 99.3%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.3%

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

      3. clear-num 99.3%

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

   5. Applied egg-rr 99.3%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.3%

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

      3. div-inv 99.3%

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

      4. associate-/r* 99.3%

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

      5. clear-num 99.3%

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

   7. Applied egg-rr 99.3%

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

   8. Taylor expanded in y around 0 74.7%

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

      1. associate-/l/ 74.7%

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

      2. cosh-def 74.7%

         \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}}}{\frac{1}{y} \cdot y} \]

      3. lft-mult-inverse 74.7%

         \[\leadsto \frac{\frac{e^{x} + e^{-x}}{2}}{\color{blue}{1}} \]

      4. associate-/l/ 74.7%

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

      5. metadata-eval 74.7%

         \[\leadsto \frac{e^{x} + e^{-x}}{\color{blue}{2}} \]

      6. expm1-log1p-u 74.7%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{x} + e^{-x}}{2}\right)\right)} \]

      7. cosh-def 74.7%

         \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cosh x}\right)\right) \]

      8. expm1-udef 74.7%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]

   10. Applied egg-rr 74.7%

       \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]
   11. Step-by-step derivation

       1. expm1-def 74.7%

          \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x\right)\right)} \]

       2. expm1-log1p 74.7%

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

   12. Simplified 74.7%

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

4. Final simplification 86.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\cosh x \leq 1.00002:\\ \;\;\;\;\frac{\sin y}{y}\\ \mathbf{else}:\\ \;\;\;\;\cosh x\\ \end{array} \]

Alternative 3: 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 99.6%

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

2. Final simplification 99.6%

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

Alternative 4: 63.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/r/ 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.1%

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

   3. clear-num 99.1%

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

5. Applied egg-rr 99.1%

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

   1. clear-num 99.1%

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

   2. associate-/r/ 99.5%

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

   3. div-inv 99.0%

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

   4. associate-/r* 99.1%

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

   5. clear-num 99.1%

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

7. Applied egg-rr 99.1%

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

8. Taylor expanded in y around 0 63.5%

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

   1. associate-/l/ 63.4%

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

   2. cosh-def 63.4%

      \[\leadsto \frac{\color{blue}{\frac{e^{x} + e^{-x}}{2}}}{\frac{1}{y} \cdot y} \]

   3. lft-mult-inverse 63.9%

      \[\leadsto \frac{\frac{e^{x} + e^{-x}}{2}}{\color{blue}{1}} \]

   4. associate-/l/ 63.9%

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

   5. metadata-eval 63.9%

      \[\leadsto \frac{e^{x} + e^{-x}}{\color{blue}{2}} \]

   6. expm1-log1p-u 63.9%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{e^{x} + e^{-x}}{2}\right)\right)} \]

   7. cosh-def 63.9%

      \[\leadsto \mathsf{expm1}\left(\mathsf{log1p}\left(\color{blue}{\cosh x}\right)\right) \]

   8. expm1-udef 63.9%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]

10. Applied egg-rr 63.9%

    \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\cosh x\right)} - 1} \]
11. Step-by-step derivation

    1. expm1-def 63.9%

       \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\cosh x\right)\right)} \]

    2. expm1-log1p 63.9%

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

12. Simplified 63.9%

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

13. Final simplification 63.9%

    \[\leadsto \cosh x \]

Alternative 5: 32.1% 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 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/r/ 99.6%

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

3. Simplified 99.6%

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

4. Taylor expanded in x around 0 48.6%

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

5. Taylor expanded in y around 0 31.2%

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

   1. *-commutative 31.2%

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

7. Simplified 31.2%

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

   1. unpow2 31.2%

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

9. Applied egg-rr 31.2%

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

10. Final simplification 31.2%

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

Alternative 6: 26.6% 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 99.6%

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

   1. *-commutative 99.6%

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

   2. associate-/r/ 99.6%

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

3. Simplified 99.6%

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

   1. clear-num 99.5%

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

   2. associate-/r/ 99.1%

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

   3. clear-num 99.1%

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

5. Applied egg-rr 99.1%

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

6. Taylor expanded in x around 0 48.1%

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

7. Taylor expanded in y around 0 26.3%

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

8. Final simplification 26.3%

   \[\leadsto 1 \]

Developer target: 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 2023310 
(FPCore (x y)
  :name "Linear.Quaternion:$csinh from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (/ (* (cosh x) (sin y)) y)

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