Linear.Quaternion:$ccosh from linear-1.19.1.3

Percentage Accurate: 89.5% → 99.8%

Time: 7.7s

Alternatives: 8

Speedup: 1.0×

• 49.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{\frac{\sin x}{x}}{\frac{1}{\sinh y}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

public static double code(double x, double y) {
	return (Math.sin(x) / x) / (1.0 / Math.sinh(y));
}

Python

def code(x, y):
	return (math.sin(x) / x) / (1.0 / math.sinh(y))

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. associate-/l* 99.5%

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

3. Simplified 99.5%

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

   1. sinh-def 67.1%

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

   2. clear-num 67.1%

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

   3. sinh-undef 99.5%

      \[\leadsto \sin x \cdot \frac{\frac{1}{\frac{2}{\color{blue}{2 \cdot \sinh y}}}}{x} \]

6. Applied egg-rr 99.5%

   \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{1}{\frac{2}{2 \cdot \sinh y}}}}{x} \]
7. Step-by-step derivation

   1. associate-/l/ 99.3%

      \[\leadsto \sin x \cdot \color{blue}{\frac{1}{x \cdot \frac{2}{2 \cdot \sinh y}}} \]

   2. associate-/r* 99.3%

      \[\leadsto \sin x \cdot \frac{1}{x \cdot \color{blue}{\frac{\frac{2}{2}}{\sinh y}}} \]

   3. metadata-eval 99.3%

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

   4. div-inv 99.3%

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

   5. div-inv 99.4%

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

   6. div-inv 99.4%

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

   7. metadata-eval 99.4%

      \[\leadsto \frac{\sin x}{x \cdot \frac{\color{blue}{\frac{2}{2}}}{\sinh y}} \]

   8. associate-/r* 99.4%

      \[\leadsto \frac{\sin x}{x \cdot \color{blue}{\frac{2}{2 \cdot \sinh y}}} \]

   9. associate-/r* 99.9%

      \[\leadsto \color{blue}{\frac{\frac{\sin x}{x}}{\frac{2}{2 \cdot \sinh y}}} \]

   10. associate-/r* 99.9%

       \[\leadsto \frac{\frac{\sin x}{x}}{\color{blue}{\frac{\frac{2}{2}}{\sinh y}}} \]

   11. metadata-eval 99.9%

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

8. Applied egg-rr 99.9%

   \[\leadsto \color{blue}{\frac{\frac{\sin x}{x}}{\frac{1}{\sinh y}}} \]
9. Add Preprocessing

Alternative 2: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-5}:\\ \;\;\;\;\frac{y}{\frac{x}{\sin x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\sinh y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 1e-5) (/ y (/ x (sin x))) (/ 1.0 (/ 1.0 (sinh y)))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 1e-5) {
		tmp = y / (x / sin(x));
	} else {
		tmp = 1.0 / (1.0 / 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) <= 1d-5) then
        tmp = y / (x / sin(x))
    else
        tmp = 1.0d0 / (1.0d0 / sinh(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 1e-5) {
		tmp = y / (x / Math.sin(x));
	} else {
		tmp = 1.0 / (1.0 / Math.sinh(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 1e-5:
		tmp = y / (x / math.sin(x))
	else:
		tmp = 1.0 / (1.0 / math.sinh(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 1e-5)
		tmp = Float64(y / Float64(x / sin(x)));
	else
		tmp = Float64(1.0 / Float64(1.0 / sinh(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 1e-5)
		tmp = y / (x / sin(x));
	else
		tmp = 1.0 / (1.0 / sinh(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-5], N[(y / N[(x / N[Sin[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 / N[(1.0 / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 1.00000000000000008e-5

   1. Initial program 83.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 58.5%

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

      1. associate-/l* 74.8%

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

   7. Simplified 74.8%

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

      1. clear-num 74.7%

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

      2. un-div-inv 74.8%

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

   9. Applied egg-rr 74.8%

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

   if 1.00000000000000008e-5 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

      1. sinh-def 98.5%

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

      2. clear-num 98.5%

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

      3. sinh-undef 98.6%

         \[\leadsto \sin x \cdot \frac{\frac{1}{\frac{2}{\color{blue}{2 \cdot \sinh y}}}}{x} \]

   6. Applied egg-rr 98.6%

      \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{1}{\frac{2}{2 \cdot \sinh y}}}}{x} \]
   7. Step-by-step derivation

      1. associate-/l/ 98.6%

         \[\leadsto \sin x \cdot \color{blue}{\frac{1}{x \cdot \frac{2}{2 \cdot \sinh y}}} \]

      2. associate-/r* 98.6%

         \[\leadsto \sin x \cdot \frac{1}{x \cdot \color{blue}{\frac{\frac{2}{2}}{\sinh y}}} \]

      3. metadata-eval 98.6%

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

      4. div-inv 98.6%

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

      5. div-inv 98.6%

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

      6. div-inv 98.6%

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

      7. metadata-eval 98.6%

         \[\leadsto \frac{\sin x}{x \cdot \frac{\color{blue}{\frac{2}{2}}}{\sinh y}} \]

      8. associate-/r* 98.6%

         \[\leadsto \frac{\sin x}{x \cdot \color{blue}{\frac{2}{2 \cdot \sinh y}}} \]

      9. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\sin x}{x}}{\frac{2}{2 \cdot \sinh y}}} \]

      10. associate-/r* 100.0%

          \[\leadsto \frac{\frac{\sin x}{x}}{\color{blue}{\frac{\frac{2}{2}}{\sinh y}}} \]

      11. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 73.9%

      \[\leadsto \frac{\color{blue}{1}}{\frac{1}{\sinh y}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 69.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\sinh y \leq 10^{-5}:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\sinh y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= (sinh y) 1e-5) (* (/ (sin x) x) y) (/ 1.0 (/ 1.0 (sinh y)))))

C

double code(double x, double y) {
	double tmp;
	if (sinh(y) <= 1e-5) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = 1.0 / (1.0 / 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) <= 1d-5) then
        tmp = (sin(x) / x) * y
    else
        tmp = 1.0d0 / (1.0d0 / sinh(y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (Math.sinh(y) <= 1e-5) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = 1.0 / (1.0 / Math.sinh(y));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if math.sinh(y) <= 1e-5:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = 1.0 / (1.0 / math.sinh(y))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (sinh(y) <= 1e-5)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(1.0 / Float64(1.0 / sinh(y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (sinh(y) <= 1e-5)
		tmp = (sin(x) / x) * y;
	else
		tmp = 1.0 / (1.0 / sinh(y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[Sinh[y], $MachinePrecision], 1e-5], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(1.0 / N[(1.0 / N[Sinh[y], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (sinh.f64 y) < 1.00000000000000008e-5

   1. Initial program 83.6%

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

      1. associate-/l* 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in y around 0 58.5%

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

      1. associate-/l* 74.8%

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

   7. Simplified 74.8%

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

   if 1.00000000000000008e-5 < (sinh.f64 y)

   1. Initial program 100.0%

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

      1. associate-/l* 98.6%

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

   3. Simplified 98.6%

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

      1. sinh-def 98.5%

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

      2. clear-num 98.5%

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

      3. sinh-undef 98.6%

         \[\leadsto \sin x \cdot \frac{\frac{1}{\frac{2}{\color{blue}{2 \cdot \sinh y}}}}{x} \]

   6. Applied egg-rr 98.6%

      \[\leadsto \sin x \cdot \frac{\color{blue}{\frac{1}{\frac{2}{2 \cdot \sinh y}}}}{x} \]
   7. Step-by-step derivation

      1. associate-/l/ 98.6%

         \[\leadsto \sin x \cdot \color{blue}{\frac{1}{x \cdot \frac{2}{2 \cdot \sinh y}}} \]

      2. associate-/r* 98.6%

         \[\leadsto \sin x \cdot \frac{1}{x \cdot \color{blue}{\frac{\frac{2}{2}}{\sinh y}}} \]

      3. metadata-eval 98.6%

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

      4. div-inv 98.6%

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

      5. div-inv 98.6%

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

      6. div-inv 98.6%

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

      7. metadata-eval 98.6%

         \[\leadsto \frac{\sin x}{x \cdot \frac{\color{blue}{\frac{2}{2}}}{\sinh y}} \]

      8. associate-/r* 98.6%

         \[\leadsto \frac{\sin x}{x \cdot \color{blue}{\frac{2}{2 \cdot \sinh y}}} \]

      9. associate-/r* 100.0%

         \[\leadsto \color{blue}{\frac{\frac{\sin x}{x}}{\frac{2}{2 \cdot \sinh y}}} \]

      10. associate-/r* 100.0%

          \[\leadsto \frac{\frac{\sin x}{x}}{\color{blue}{\frac{\frac{2}{2}}{\sinh y}}} \]

      11. metadata-eval 100.0%

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

   8. Applied egg-rr 100.0%

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

   9. Taylor expanded in x around 0 73.9%

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

4. Final simplification 74.5%

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

Alternative 4: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. associate-/l* 99.5%

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

3. Simplified 99.5%

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

Alternative 5: 54.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 3900000:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= y 3900000.0) (* (/ (sin x) x) y) (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 3900000.0) {
		tmp = (sin(x) / x) * y;
	} else {
		tmp = (x * y) / 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 <= 3900000.0d0) then
        tmp = (sin(x) / x) * y
    else
        tmp = (x * y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 3900000.0) {
		tmp = (Math.sin(x) / x) * y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 3900000.0:
		tmp = (math.sin(x) / x) * y
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 3900000.0)
		tmp = Float64(Float64(sin(x) / x) * y);
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 3900000.0)
		tmp = (sin(x) / x) * y;
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 3900000.0], N[(N[(N[Sin[x], $MachinePrecision] / x), $MachinePrecision] * y), $MachinePrecision], N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 3.9e6

   1. Initial program 83.9%

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

      1. associate-/l* 99.4%

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

   3. Simplified 99.4%

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

   5. Taylor expanded in y around 0 57.8%

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

      1. associate-/l* 73.8%

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

   7. Simplified 73.8%

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

   if 3.9e6 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0 4.6%

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

   4. Taylor expanded in x around 0 15.6%

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

      1. *-commutative 15.6%

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

   6. Simplified 15.6%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 3900000:\\ \;\;\;\;\frac{\sin x}{x} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 88.0%

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

   1. associate-/l* 99.5%

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

3. Simplified 99.5%

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

5. Taylor expanded in y around 0 70.9%

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

Alternative 7: 31.0% accurate, 20.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+22}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= x 4e+22) y (/ (* x y) x)))

C

double code(double x, double y) {
	double tmp;
	if (x <= 4e+22) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= 4d+22) then
        tmp = y
    else
        tmp = (x * y) / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= 4e+22) {
		tmp = y;
	} else {
		tmp = (x * y) / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= 4e+22:
		tmp = y
	else:
		tmp = (x * y) / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= 4e+22)
		tmp = y;
	else
		tmp = Float64(Float64(x * y) / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= 4e+22)
		tmp = y;
	else
		tmp = (x * y) / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, 4e+22], y, N[(N[(x * y), $MachinePrecision] / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 4e22

   1. Initial program 84.5%

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

   3. Taylor expanded in y around 0 38.7%

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

      1. associate-*l/ 54.1%

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

      2. div-inv 54.1%

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

      3. associate-*l* 73.4%

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

   5. Applied egg-rr 73.4%

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

   6. Taylor expanded in x around 0 37.1%

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

   if 4e22 < x

   1. Initial program 99.8%

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

   3. Taylor expanded in y around 0 62.0%

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

   4. Taylor expanded in x around 0 16.0%

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

      1. *-commutative 16.0%

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

   6. Simplified 16.0%

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

4. Final simplification 32.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4 \cdot 10^{+22}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 28.1% accurate, 205.0× speedup

Math

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

FPCore

(FPCore (x y) :precision binary64 y)

C

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

Fortran

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

Java

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

Python

def code(x, y):
	return y

Julia

function code(x, y)
	return y
end

MATLAB

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

Wolfram

code[x_, y_] := y

Derivation

1. Initial program 88.0%

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

3. Taylor expanded in y around 0 44.1%

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

   1. associate-*l/ 56.0%

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

   2. div-inv 55.9%

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

   3. associate-*l* 70.8%

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

5. Applied egg-rr 70.8%

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

6. Taylor expanded in x around 0 29.7%

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

Developer target: 99.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024106 
(FPCore (x y)
  :name "Linear.Quaternion:$ccosh from linear-1.19.1.3"
  :precision binary64

  :alt
  (* (sin x) (/ (sinh y) x))

  (/ (* (sin x) (sinh y)) x))