sintan (problem 3.4.5)

Percentage Accurate: 52.0% → 99.9%

Time: 11.0s

Alternatives: 5

Speedup: 67.6×

Specification

Math

\[\begin{array}{l} \\ \frac{x - \sin x}{x - \tan x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))

C

double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x - sin(x)) / (x - tan(x))
end function

Java

public static double code(double x) {
	return (x - Math.sin(x)) / (x - Math.tan(x));
}

Python

def code(x):
	return (x - math.sin(x)) / (x - math.tan(x))

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
end

MATLAB

function tmp = code(x)
	tmp = (x - sin(x)) / (x - tan(x));
end

Wolfram

code[x_] := N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 52.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - \sin x}{x - \tan x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))

C

double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x - sin(x)) / (x - tan(x))
end function

Java

public static double code(double x) {
	return (x - Math.sin(x)) / (x - Math.tan(x));
}

Python

def code(x):
	return (x - math.sin(x)) / (x - math.tan(x))

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
end

MATLAB

function tmp = code(x)
	tmp = (x - sin(x)) / (x - tan(x));
end

Wolfram

code[x_] := N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0054:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \sin x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0054)
   (- (* 0.225 (pow x_m 2.0)) 0.5)
   (/ (- x_m (sin x_m)) (- x_m (tan x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.0054) {
		tmp = (0.225 * pow(x_m, 2.0)) - 0.5;
	} else {
		tmp = (x_m - sin(x_m)) / (x_m - tan(x_m));
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 0.0054d0) then
        tmp = (0.225d0 * (x_m ** 2.0d0)) - 0.5d0
    else
        tmp = (x_m - sin(x_m)) / (x_m - tan(x_m))
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.0054) {
		tmp = (0.225 * Math.pow(x_m, 2.0)) - 0.5;
	} else {
		tmp = (x_m - Math.sin(x_m)) / (x_m - Math.tan(x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.0054:
		tmp = (0.225 * math.pow(x_m, 2.0)) - 0.5
	else:
		tmp = (x_m - math.sin(x_m)) / (x_m - math.tan(x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.0054)
		tmp = Float64(Float64(0.225 * (x_m ^ 2.0)) - 0.5);
	else
		tmp = Float64(Float64(x_m - sin(x_m)) / Float64(x_m - tan(x_m)));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.0054)
		tmp = (0.225 * (x_m ^ 2.0)) - 0.5;
	else
		tmp = (x_m - sin(x_m)) / (x_m - tan(x_m));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.0054], N[(N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[(x$95$m - N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m - N[Tan[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0054000000000000003

   1. Initial program 38.1%

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Taylor expanded in x around 0 63.5%

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]

   if 0.0054000000000000003 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0054:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 2: 98.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.3:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \left|\sin x_m\right|}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.3)
   (- (* 0.225 (pow x_m 2.0)) 0.5)
   (/ (- x_m (fabs (sin x_m))) x_m)))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.3) {
		tmp = (0.225 * pow(x_m, 2.0)) - 0.5;
	} else {
		tmp = (x_m - fabs(sin(x_m))) / x_m;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.3d0) then
        tmp = (0.225d0 * (x_m ** 2.0d0)) - 0.5d0
    else
        tmp = (x_m - abs(sin(x_m))) / x_m
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.3) {
		tmp = (0.225 * Math.pow(x_m, 2.0)) - 0.5;
	} else {
		tmp = (x_m - Math.abs(Math.sin(x_m))) / x_m;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.3:
		tmp = (0.225 * math.pow(x_m, 2.0)) - 0.5
	else:
		tmp = (x_m - math.fabs(math.sin(x_m))) / x_m
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.3)
		tmp = Float64(Float64(0.225 * (x_m ^ 2.0)) - 0.5);
	else
		tmp = Float64(Float64(x_m - abs(sin(x_m))) / x_m);
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.3)
		tmp = (0.225 * (x_m ^ 2.0)) - 0.5;
	else
		tmp = (x_m - abs(sin(x_m))) / x_m;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.3], N[(N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[(x$95$m - N[Abs[N[Sin[x$95$m], $MachinePrecision]], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 38.1%

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Taylor expanded in x around 0 63.5%

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]

   if 2.2999999999999998 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Taylor expanded in x around inf 100.0%

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

      1. add-sqr-sqrt 50.0%

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

      2. sqrt-unprod 100.0%

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

      3. pow2 100.0%

         \[\leadsto \frac{x - \sqrt{\color{blue}{{\sin x}^{2}}}}{x} \]

   4. Applied egg-rr 100.0%

      \[\leadsto \frac{x - \color{blue}{\sqrt{{\sin x}^{2}}}}{x} \]
   5. Step-by-step derivation

      1. unpow2 100.0%

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

      2. rem-sqrt-square 100.0%

         \[\leadsto \frac{x - \color{blue}{\left|\sin x\right|}}{x} \]

   6. Simplified 100.0%

      \[\leadsto \frac{x - \color{blue}{\left|\sin x\right|}}{x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.3:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \left|\sin x\right|}{x}\\ \end{array} \]

Alternative 3: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.55:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.55) (- (* 0.225 (pow x_m 2.0)) 0.5) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.55) {
		tmp = (0.225 * pow(x_m, 2.0)) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.55d0) then
        tmp = (0.225d0 * (x_m ** 2.0d0)) - 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.55) {
		tmp = (0.225 * Math.pow(x_m, 2.0)) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.55:
		tmp = (0.225 * math.pow(x_m, 2.0)) - 0.5
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.55)
		tmp = Float64(Float64(0.225 * (x_m ^ 2.0)) - 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.55)
		tmp = (0.225 * (x_m ^ 2.0)) - 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.55], N[(N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.5499999999999998

   1. Initial program 38.1%

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Taylor expanded in x around 0 63.5%

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]

   if 2.5499999999999998 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.55:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 4: 98.4% accurate, 67.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.56:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.56) -0.5 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.56) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 1.56d0) then
        tmp = -0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 1.56) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.56:
		tmp = -0.5
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.56)
		tmp = -0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 1.56)
		tmp = -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 1.56], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.5600000000000001

   1. Initial program 38.1%

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Taylor expanded in x around 0 62.5%

      \[\leadsto \color{blue}{-0.5} \]

   if 1.5600000000000001 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]

   2. Taylor expanded in x around inf 100.0%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 49.2% accurate, 207.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ -0.5 \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 -0.5)

C

x_m = fabs(x);
double code(double x_m) {
	return -0.5;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    code = -0.5d0
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	return -0.5;
}

Python

x_m = math.fabs(x)
def code(x_m):
	return -0.5

Julia

x_m = abs(x)
function code(x_m)
	return -0.5
end

MATLAB

x_m = abs(x);
function tmp = code(x_m)
	tmp = -0.5;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := -0.5

Derivation

1. Initial program 55.5%

   \[\frac{x - \sin x}{x - \tan x} \]

2. Taylor expanded in x around 0 45.4%

   \[\leadsto \color{blue}{-0.5} \]

3. Final simplification 45.4%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023318 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))