Linear.Quaternion:$ccos from linear-1.19.1.3

Percentage Accurate: 99.9% → 99.9%

Time: 8.5s

Alternatives: 10

Speedup: 1.0×

• Could not converge on a ground truth

  1. reg0 = (bf "2.411855566653070146050783712392571989933e268")
  2. reg1 = (bf "1.610772159187373647937101677285333312033e168")

• 49.3% 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: 99.9% 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: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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 77.0%

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

   2. *-un-lft-identity 77.0%

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

   3. log-prod 77.0%

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

   4. metadata-eval 77.0%

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

   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/ 87.1%

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

   3. associate-*l/ 88.5%

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

   4. associate-/r/ 100.0%

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

6. Simplified 100.0%

   \[\leadsto \color{blue}{\frac{\sin x}{\frac{y}{\sinh y}}} \]
7. Add Preprocessing

Alternative 2: 87.6% 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 100.0%

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

   3. Taylor expanded in x around 0 75.8%

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

      1. add-log-exp 73.0%

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

      2. *-un-lft-identity 73.0%

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

      3. log-prod 73.0%

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

      4. metadata-eval 73.0%

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

      5. add-log-exp 75.8%

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

   5. Applied egg-rr 75.8%

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

      1. +-lft-identity 75.8%

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

      2. associate-*r/ 75.8%

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

   7. Simplified 75.8%

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

Alternative 3: 87.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(sinh(y) / y) <= 1.0)
		tmp = sin(x);
	else
		tmp = Float64(x / Float64(y / sinh(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 / (y / sinh(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[(x / N[(y / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $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 100.0%

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

   3. Taylor expanded in x around 0 75.8%

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

      1. clear-num 75.8%

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

      2. un-div-inv 75.8%

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

   5. Applied egg-rr 75.8%

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

Alternative 4: 87.6% 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 100.0%

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

   3. Taylor expanded in x around 0 75.8%

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

Alternative 5: 99.9% 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 6: 55.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 580:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x \cdot \left(x \cdot y\right)}{y} \cdot -0.16666666666666666\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 580.0)
   (sin x)
   (* x (+ 1.0 (* (/ (* x (* x y)) y) -0.16666666666666666)))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 580.0) {
		tmp = sin(x);
	} else {
		tmp = x * (1.0 + (((x * (x * 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 <= 580.0d0) then
        tmp = sin(x)
    else
        tmp = x * (1.0d0 + (((x * (x * y)) / y) * (-0.16666666666666666d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 580.0) {
		tmp = Math.sin(x);
	} else {
		tmp = x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 580.0:
		tmp = math.sin(x)
	else:
		tmp = x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 580.0)
		tmp = sin(x);
	else
		tmp = Float64(x * Float64(1.0 + Float64(Float64(Float64(x * Float64(x * y)) / y) * -0.16666666666666666)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 580.0)
		tmp = sin(x);
	else
		tmp = x * (1.0 + (((x * (x * y)) / y) * -0.16666666666666666));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 580.0], N[Sin[x], $MachinePrecision], N[(x * N[(1.0 + N[(N[(N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 580

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 69.8%

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

   if 580 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 2.8%

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

   4. Taylor expanded in x around 0 21.7%

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

      1. *-commutative 21.7%

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

   6. Simplified 21.7%

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

      1. unpow2 21.7%

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

      2. *-un-lft-identity 21.7%

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

      3. associate-*r* 21.7%

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

      4. *-inverses 21.7%

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

      5. div-inv 21.7%

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

      6. associate-*l* 21.7%

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

      7. div-inv 21.7%

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

      8. associate-*l/ 22.9%

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

      9. *-commutative 22.9%

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

   8. Applied egg-rr 22.9%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 580:\\ \;\;\;\;\sin x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(1 + \frac{x \cdot \left(x \cdot y\right)}{y} \cdot -0.16666666666666666\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 34.2% accurate, 14.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+181}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 4.5e+181)
   (* x (+ 1.0 (* -0.16666666666666666 (* x x))))
   (* (* x y) (/ 1.0 y))))

C

double code(double x, double y) {
	double tmp;
	if (y <= 4.5e+181) {
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)));
	} else {
		tmp = (x * y) * (1.0 / 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 <= 4.5d+181) then
        tmp = x * (1.0d0 + ((-0.16666666666666666d0) * (x * x)))
    else
        tmp = (x * y) * (1.0d0 / y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 4.5e+181) {
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)));
	} else {
		tmp = (x * y) * (1.0 / y);
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 4.5e+181:
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)))
	else:
		tmp = (x * y) * (1.0 / y)
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 4.5e+181)
		tmp = Float64(x * Float64(1.0 + Float64(-0.16666666666666666 * Float64(x * x))));
	else
		tmp = Float64(Float64(x * y) * Float64(1.0 / y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 4.5e+181)
		tmp = x * (1.0 + (-0.16666666666666666 * (x * x)));
	else
		tmp = (x * y) * (1.0 / y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 4.5e+181], N[(x * N[(1.0 + N[(-0.16666666666666666 * N[(x * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x * y), $MachinePrecision] * N[(1.0 / y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 4.5e181

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 55.2%

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

   4. Taylor expanded in x around 0 34.8%

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

      1. *-commutative 34.8%

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

   6. Simplified 34.8%

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

      1. unpow2 34.8%

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

   8. Applied egg-rr 34.8%

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

   if 4.5e181 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 82.6%

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

      1. add-log-exp 82.6%

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

      2. *-un-lft-identity 82.6%

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

      3. log-prod 82.6%

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

      4. metadata-eval 82.6%

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

      5. add-log-exp 82.6%

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

   5. Applied egg-rr 82.6%

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

      1. +-lft-identity 82.6%

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

      2. associate-*r/ 82.6%

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

   7. Simplified 82.6%

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

   8. Taylor expanded in y around 0 31.8%

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

      1. clear-num 31.8%

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

      2. associate-/r/ 31.8%

         \[\leadsto \color{blue}{\frac{1}{y} \cdot \left(x \cdot y\right)} \]

      3. *-commutative 31.8%

         \[\leadsto \frac{1}{y} \cdot \color{blue}{\left(y \cdot x\right)} \]

   10. Applied egg-rr 31.8%

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

4. Final simplification 34.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 4.5 \cdot 10^{+181}:\\ \;\;\;\;x \cdot \left(1 + -0.16666666666666666 \cdot \left(x \cdot x\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x \cdot y\right) \cdot \frac{1}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 34.6% accurate, 15.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(x * N[(1.0 + N[(N[(N[(x * N[(x * y), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision] * -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 50.5%

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

4. Taylor expanded in x around 0 33.0%

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

   1. *-commutative 33.0%

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

6. Simplified 33.0%

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

   1. unpow2 33.0%

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

   2. *-un-lft-identity 33.0%

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

   3. associate-*r* 33.0%

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

   4. *-inverses 33.0%

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

   5. div-inv 33.0%

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

   6. associate-*l* 33.7%

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

   7. div-inv 33.7%

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

   8. associate-*l/ 34.4%

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

   9. *-commutative 34.4%

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

8. Applied egg-rr 34.4%

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

9. Final simplification 34.4%

   \[\leadsto x \cdot \left(1 + \frac{x \cdot \left(x \cdot y\right)}{y} \cdot -0.16666666666666666\right) \]
10. Add Preprocessing

Alternative 9: 28.9% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 5.6 \cdot 10^{+113}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 5.6e+113) x (/ (* x y) y)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 5.6e+113) {
		tmp = x;
	} else {
		tmp = (x * 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 (x <= 5.6d+113) then
        tmp = x
    else
        tmp = (x * y) / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 5.6e+113) {
		tmp = x;
	} else {
		tmp = (x * y) / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 5.6e+113:
		tmp = x
	else:
		tmp = (x * y) / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 5.6e+113)
		tmp = x;
	else
		tmp = Float64(Float64(x * y) / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 5.6e+113)
		tmp = x;
	else
		tmp = (x * y) / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 5.6e+113], x, N[(N[(x * y), $MachinePrecision] / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 5.59999999999999995e113

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 69.1%

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

   4. Taylor expanded in y around 0 30.0%

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

   if 5.59999999999999995e113 < x

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 29.8%

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

      1. add-log-exp 29.5%

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

      2. *-un-lft-identity 29.5%

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

      3. log-prod 29.5%

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

      4. metadata-eval 29.5%

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

      5. add-log-exp 29.8%

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

   5. Applied egg-rr 29.8%

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

      1. +-lft-identity 29.8%

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

      2. associate-*r/ 29.8%

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

   7. Simplified 29.8%

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

   8. Taylor expanded in y around 0 14.1%

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

Alternative 10: 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 62.6%

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

4. Taylor expanded in y around 0 25.5%

   \[\leadsto \color{blue}{x} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024154 
(FPCore (x y)
  :name "Linear.Quaternion:$ccos from linear-1.19.1.3"
  :precision binary64
  (* (sin x) (/ (sinh y) y)))