Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 100.0% → 100.0%

Time: 6.8s

Alternatives: 7

Speedup: 1.0×

• 49.6% 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. Final simplification 100.0%

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

Alternative 2: 86.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{\sinh y}{y}\\ \mathbf{if}\;t_0 \leq 1.01:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot t_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (sinh y) y))) (if (<= t_0 1.01) (sin x) (* x t_0))))

C

double code(double x, double y) {
	double t_0 = sinh(y) / y;
	double tmp;
	if (t_0 <= 1.01) {
		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 <= 1.01d0) 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 <= 1.01) {
		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 <= 1.01:
		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 <= 1.01)
		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 <= 1.01)
		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, 1.01], N[Sin[x], $MachinePrecision], N[(x * t$95$0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (sinh.f64 y) y) < 1.01000000000000001

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 99.8%

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

   if 1.01000000000000001 < (/.f64 (sinh.f64 y) y)

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 65.9%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\sinh y}{y} \leq 1.01:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{\sinh y}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 53.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+57}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e+57) (sin x) (* (/ 1.0 y) (/ y (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e+57) {
		tmp = sin(x);
	} else {
		tmp = (1.0 / y) * (y / (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9d+57) then
        tmp = sin(x)
    else
        tmp = (1.0d0 / y) * (y / (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e+57) {
		tmp = Math.sin(x);
	} else {
		tmp = (1.0 / y) * (y / (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e+57:
		tmp = math.sin(x)
	else:
		tmp = (1.0 / y) * (y / (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e+57)
		tmp = sin(x);
	else
		tmp = Float64(Float64(1.0 / y) * Float64(y / Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e+57)
		tmp = sin(x);
	else
		tmp = (1.0 / y) * (y / (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e+57], N[Sin[x], $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(y / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.99999999999999991e57

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 64.2%

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

   if 8.99999999999999991e57 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. clear-num 100.0%

         \[\leadsto 0 + \sin x \cdot \color{blue}{\frac{1}{\frac{y}{\sinh y}}} \]

      7. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 69.1%

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

   6. Simplified 69.1%

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

   7. Taylor expanded in y around 0 2.5%

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

   8. Taylor expanded in x around 0 2.5%

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

      1. *-un-lft-identity 2.5%

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

      2. div-inv 2.5%

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

      3. times-frac 16.2%

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

   10. Applied egg-rr 16.2%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{+57}:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 29.4% accurate, 12.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.34:\\ \;\;\;\;\frac{y}{0.16666666666666666 \cdot \left(x \cdot y\right) + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.34)
   (/ y (+ (* 0.16666666666666666 (* x y)) (/ y x)))
   (* (/ 1.0 y) (/ y (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.34) {
		tmp = y / ((0.16666666666666666 * (x * y)) + (y / x));
	} else {
		tmp = (1.0 / y) * (y / (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.34d0) then
        tmp = y / ((0.16666666666666666d0 * (x * y)) + (y / x))
    else
        tmp = (1.0d0 / y) * (y / (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.34) {
		tmp = y / ((0.16666666666666666 * (x * y)) + (y / x));
	} else {
		tmp = (1.0 / y) * (y / (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.34:
		tmp = y / ((0.16666666666666666 * (x * y)) + (y / x))
	else:
		tmp = (1.0 / y) * (y / (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.34)
		tmp = Float64(y / Float64(Float64(0.16666666666666666 * Float64(x * y)) + Float64(y / x)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(y / Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.34)
		tmp = y / ((0.16666666666666666 * (x * y)) + (y / x));
	else
		tmp = (1.0 / y) * (y / (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.34], N[(y / N[(N[(0.16666666666666666 * N[(x * y), $MachinePrecision]), $MachinePrecision] + N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(y / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.340000000000000024

   1. Initial program 100.0%

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

      1. add-log-exp 64.5%

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

      2. *-un-lft-identity 64.5%

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

      3. log-prod 64.5%

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

      4. metadata-eval 64.5%

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

      5. add-log-exp 100.0%

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

      6. clear-num 100.0%

         \[\leadsto 0 + \sin x \cdot \color{blue}{\frac{1}{\frac{y}{\sinh y}}} \]

      7. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-/l* 79.5%

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

      3. *-commutative 79.5%

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

      4. associate-/l* 90.5%

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

   6. Simplified 90.5%

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

   7. Taylor expanded in y around 0 66.5%

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

   8. Taylor expanded in x around 0 38.8%

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

   if 0.340000000000000024 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. clear-num 100.0%

         \[\leadsto 0 + \sin x \cdot \color{blue}{\frac{1}{\frac{y}{\sinh y}}} \]

      7. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in y around 0 2.6%

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

   8. Taylor expanded in x around 0 2.4%

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

      1. *-un-lft-identity 2.4%

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

      2. div-inv 2.4%

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

      3. times-frac 14.4%

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

   10. Applied egg-rr 14.4%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.34:\\ \;\;\;\;\frac{y}{0.16666666666666666 \cdot \left(x \cdot y\right) + \frac{y}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 29.3% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 9e-8) x (* (/ 1.0 y) (/ y (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 9e-8) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (y / (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 9d-8) then
        tmp = x
    else
        tmp = (1.0d0 / y) * (y / (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 9e-8) {
		tmp = x;
	} else {
		tmp = (1.0 / y) * (y / (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 9e-8:
		tmp = x
	else:
		tmp = (1.0 / y) * (y / (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 9e-8)
		tmp = x;
	else
		tmp = Float64(Float64(1.0 / y) * Float64(y / Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 9e-8)
		tmp = x;
	else
		tmp = (1.0 / y) * (y / (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 9e-8], x, N[(N[(1.0 / y), $MachinePrecision] * N[(y / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 8.99999999999999986e-8

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 58.5%

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

   4. Taylor expanded in y around 0 38.7%

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

   if 8.99999999999999986e-8 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. clear-num 100.0%

         \[\leadsto 0 + \sin x \cdot \color{blue}{\frac{1}{\frac{y}{\sinh y}}} \]

      7. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in y around 0 2.6%

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

   8. Taylor expanded in x around 0 2.4%

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

      1. *-un-lft-identity 2.4%

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

      2. div-inv 2.4%

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

      3. times-frac 14.4%

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

   10. Applied egg-rr 14.4%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 9 \cdot 10^{-8}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 29.4% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 0.34:\\ \;\;\;\;\frac{1}{\frac{1}{x} + x \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 0.34)
   (/ 1.0 (+ (/ 1.0 x) (* x 0.16666666666666666)))
   (* (/ 1.0 y) (/ y (/ 1.0 x)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 0.34) {
		tmp = 1.0 / ((1.0 / x) + (x * 0.16666666666666666));
	} else {
		tmp = (1.0 / y) * (y / (1.0 / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 0.34d0) then
        tmp = 1.0d0 / ((1.0d0 / x) + (x * 0.16666666666666666d0))
    else
        tmp = (1.0d0 / y) * (y / (1.0d0 / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 0.34) {
		tmp = 1.0 / ((1.0 / x) + (x * 0.16666666666666666));
	} else {
		tmp = (1.0 / y) * (y / (1.0 / x));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 0.34:
		tmp = 1.0 / ((1.0 / x) + (x * 0.16666666666666666))
	else:
		tmp = (1.0 / y) * (y / (1.0 / x))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 0.34)
		tmp = Float64(1.0 / Float64(Float64(1.0 / x) + Float64(x * 0.16666666666666666)));
	else
		tmp = Float64(Float64(1.0 / y) * Float64(y / Float64(1.0 / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 0.34)
		tmp = 1.0 / ((1.0 / x) + (x * 0.16666666666666666));
	else
		tmp = (1.0 / y) * (y / (1.0 / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 0.34], N[(1.0 / N[(N[(1.0 / x), $MachinePrecision] + N[(x * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(1.0 / y), $MachinePrecision] * N[(y / N[(1.0 / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 0.340000000000000024

   1. Initial program 100.0%

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

      1. add-log-exp 64.5%

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

      2. *-un-lft-identity 64.5%

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

      3. log-prod 64.5%

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

      4. metadata-eval 64.5%

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

      5. add-log-exp 100.0%

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

      6. clear-num 100.0%

         \[\leadsto 0 + \sin x \cdot \color{blue}{\frac{1}{\frac{y}{\sinh y}}} \]

      7. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-/l* 79.5%

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

      3. *-commutative 79.5%

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

      4. associate-/l* 90.5%

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

   6. Simplified 90.5%

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

   7. Taylor expanded in y around 0 66.5%

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

   8. Taylor expanded in x around 0 38.8%

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

   9. Taylor expanded in y around 0 38.8%

      \[\leadsto \color{blue}{\frac{1}{0.16666666666666666 \cdot x + \frac{1}{x}}} \]

   if 0.340000000000000024 < y

   1. Initial program 100.0%

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

      1. add-log-exp 100.0%

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

      2. *-un-lft-identity 100.0%

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

      3. log-prod 100.0%

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

      4. metadata-eval 100.0%

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

      5. add-log-exp 100.0%

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

      6. clear-num 100.0%

         \[\leadsto 0 + \sin x \cdot \color{blue}{\frac{1}{\frac{y}{\sinh y}}} \]

      7. un-div-inv 100.0%

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

   4. Applied egg-rr 100.0%

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

      1. +-lft-identity 100.0%

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

      2. associate-/l* 100.0%

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

      3. *-commutative 100.0%

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

      4. associate-/l* 73.0%

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

   6. Simplified 73.0%

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

   7. Taylor expanded in y around 0 2.6%

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

   8. Taylor expanded in x around 0 2.4%

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

      1. *-un-lft-identity 2.4%

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

      2. div-inv 2.4%

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

      3. times-frac 14.4%

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

   10. Applied egg-rr 14.4%

       \[\leadsto \color{blue}{\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 32.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 0.34:\\ \;\;\;\;\frac{1}{\frac{1}{x} + x \cdot 0.16666666666666666}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{y} \cdot \frac{y}{\frac{1}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 26.5% 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. Taylor expanded in x around 0 61.7%

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

4. Taylor expanded in y around 0 29.8%

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

5. Final simplification 29.8%

   \[\leadsto x \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024019 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))