sintan (problem 3.4.5)

Percentage Accurate: 51.0% → 100.0%

Time: 15.4s

Alternatives: 8

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: 51.0% 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: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := x_m - \tan x_m\\ \mathbf{if}\;x_m \leq 0.098:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x_m}^{4} + \left(0.00024107142857142857 \cdot {x_m}^{6} + 0.225 \cdot {x_m}^{2}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{t_0} - \frac{\sin x_m}{t_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- x_m (tan x_m))))
   (if (<= x_m 0.098)
     (-
      (+
       (* -0.009642857142857142 (pow x_m 4.0))
       (+ (* 0.00024107142857142857 (pow x_m 6.0)) (* 0.225 (pow x_m 2.0))))
      0.5)
     (- (/ x_m t_0) (/ (sin x_m) t_0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = x_m - tan(x_m);
	double tmp;
	if (x_m <= 0.098) {
		tmp = ((-0.009642857142857142 * pow(x_m, 4.0)) + ((0.00024107142857142857 * pow(x_m, 6.0)) + (0.225 * pow(x_m, 2.0)))) - 0.5;
	} else {
		tmp = (x_m / t_0) - (sin(x_m) / t_0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m - tan(x_m)
    if (x_m <= 0.098d0) then
        tmp = (((-0.009642857142857142d0) * (x_m ** 4.0d0)) + ((0.00024107142857142857d0 * (x_m ** 6.0d0)) + (0.225d0 * (x_m ** 2.0d0)))) - 0.5d0
    else
        tmp = (x_m / t_0) - (sin(x_m) / t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = x_m - Math.tan(x_m);
	double tmp;
	if (x_m <= 0.098) {
		tmp = ((-0.009642857142857142 * Math.pow(x_m, 4.0)) + ((0.00024107142857142857 * Math.pow(x_m, 6.0)) + (0.225 * Math.pow(x_m, 2.0)))) - 0.5;
	} else {
		tmp = (x_m / t_0) - (Math.sin(x_m) / t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = x_m - math.tan(x_m)
	tmp = 0
	if x_m <= 0.098:
		tmp = ((-0.009642857142857142 * math.pow(x_m, 4.0)) + ((0.00024107142857142857 * math.pow(x_m, 6.0)) + (0.225 * math.pow(x_m, 2.0)))) - 0.5
	else:
		tmp = (x_m / t_0) - (math.sin(x_m) / t_0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(x_m - tan(x_m))
	tmp = 0.0
	if (x_m <= 0.098)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x_m ^ 4.0)) + Float64(Float64(0.00024107142857142857 * (x_m ^ 6.0)) + Float64(0.225 * (x_m ^ 2.0)))) - 0.5);
	else
		tmp = Float64(Float64(x_m / t_0) - Float64(sin(x_m) / t_0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = x_m - tan(x_m);
	tmp = 0.0;
	if (x_m <= 0.098)
		tmp = ((-0.009642857142857142 * (x_m ^ 4.0)) + ((0.00024107142857142857 * (x_m ^ 6.0)) + (0.225 * (x_m ^ 2.0)))) - 0.5;
	else
		tmp = (x_m / t_0) - (sin(x_m) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(x$95$m - N[Tan[x$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.098], N[(N[(N[(-0.009642857142857142 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.00024107142857142857 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[(x$95$m / t$95$0), $MachinePrecision] - N[(N[Sin[x$95$m], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.098000000000000004

   1. Initial program 33.3%

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

   2. Taylor expanded in x around 0 69.6%

      \[\leadsto \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + 0.225 \cdot {x}^{2}\right)\right) - 0.5} \]

   if 0.098000000000000004 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.098:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + 0.225 \cdot {x}^{2}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} t_0 := x_m - \tan x_m\\ \mathbf{if}\;x_m \leq 0.033:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x_m}^{4} + 0.225 \cdot {x_m}^{2}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{t_0} - \frac{\sin x_m}{t_0}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (let* ((t_0 (- x_m (tan x_m))))
   (if (<= x_m 0.033)
     (-
      (+ (* -0.009642857142857142 (pow x_m 4.0)) (* 0.225 (pow x_m 2.0)))
      0.5)
     (- (/ x_m t_0) (/ (sin x_m) t_0)))))

C

x_m = fabs(x);
double code(double x_m) {
	double t_0 = x_m - tan(x_m);
	double tmp;
	if (x_m <= 0.033) {
		tmp = ((-0.009642857142857142 * pow(x_m, 4.0)) + (0.225 * pow(x_m, 2.0))) - 0.5;
	} else {
		tmp = (x_m / t_0) - (sin(x_m) / t_0);
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x_m - tan(x_m)
    if (x_m <= 0.033d0) then
        tmp = (((-0.009642857142857142d0) * (x_m ** 4.0d0)) + (0.225d0 * (x_m ** 2.0d0))) - 0.5d0
    else
        tmp = (x_m / t_0) - (sin(x_m) / t_0)
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double t_0 = x_m - Math.tan(x_m);
	double tmp;
	if (x_m <= 0.033) {
		tmp = ((-0.009642857142857142 * Math.pow(x_m, 4.0)) + (0.225 * Math.pow(x_m, 2.0))) - 0.5;
	} else {
		tmp = (x_m / t_0) - (Math.sin(x_m) / t_0);
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	t_0 = x_m - math.tan(x_m)
	tmp = 0
	if x_m <= 0.033:
		tmp = ((-0.009642857142857142 * math.pow(x_m, 4.0)) + (0.225 * math.pow(x_m, 2.0))) - 0.5
	else:
		tmp = (x_m / t_0) - (math.sin(x_m) / t_0)
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	t_0 = Float64(x_m - tan(x_m))
	tmp = 0.0
	if (x_m <= 0.033)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x_m ^ 4.0)) + Float64(0.225 * (x_m ^ 2.0))) - 0.5);
	else
		tmp = Float64(Float64(x_m / t_0) - Float64(sin(x_m) / t_0));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	t_0 = x_m - tan(x_m);
	tmp = 0.0;
	if (x_m <= 0.033)
		tmp = ((-0.009642857142857142 * (x_m ^ 4.0)) + (0.225 * (x_m ^ 2.0))) - 0.5;
	else
		tmp = (x_m / t_0) - (sin(x_m) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := Block[{t$95$0 = N[(x$95$m - N[Tan[x$95$m], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x$95$m, 0.033], N[(N[(N[(-0.009642857142857142 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[(x$95$m / t$95$0), $MachinePrecision] - N[(N[Sin[x$95$m], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.033000000000000002

   1. Initial program 33.3%

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

   2. Taylor expanded in x around 0 69.2%

      \[\leadsto \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot {x}^{2}\right) - 0.5} \]

   if 0.033000000000000002 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Step-by-step derivation

      1. div-sub 100.0%

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

   3. Applied egg-rr 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.033:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot {x}^{2}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array} \]

Alternative 3: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0295:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x_m}^{4} + 0.225 \cdot {x_m}^{2}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \sin x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0295)
   (- (+ (* -0.009642857142857142 (pow x_m 4.0)) (* 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 <= 0.0295) {
		tmp = ((-0.009642857142857142 * pow(x_m, 4.0)) + (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 <= 0.0295d0) then
        tmp = (((-0.009642857142857142d0) * (x_m ** 4.0d0)) + (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 <= 0.0295) {
		tmp = ((-0.009642857142857142 * Math.pow(x_m, 4.0)) + (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 <= 0.0295:
		tmp = ((-0.009642857142857142 * math.pow(x_m, 4.0)) + (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 <= 0.0295)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x_m ^ 4.0)) + 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 <= 0.0295)
		tmp = ((-0.009642857142857142 * (x_m ^ 4.0)) + (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, 0.0295], N[(N[(N[(-0.009642857142857142 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $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 < 0.029499999999999998

   1. Initial program 33.3%

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

   2. Taylor expanded in x around 0 69.2%

      \[\leadsto \color{blue}{\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot {x}^{2}\right) - 0.5} \]

   if 0.029499999999999998 < x

   1. Initial program 100.0%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0295:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot {x}^{2}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 4: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.005:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \sin x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.005)
   (- (* 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 <= 0.005) {
		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 <= 0.005d0) 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 <= 0.005) {
		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 <= 0.005:
		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 <= 0.005)
		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 <= 0.005)
		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, 0.005], 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 < 0.0050000000000000001

   1. Initial program 33.3%

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

   2. Taylor expanded in x around 0 70.1%

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

   if 0.0050000000000000001 < x

   1. Initial program 100.0%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.9× speedup

Math

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

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 (- x_m (tan 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 / (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 <= 2.3d0) then
        tmp = (0.225d0 * (x_m ** 2.0d0)) - 0.5d0
    else
        tmp = 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 <= 2.3) {
		tmp = (0.225 * Math.pow(x_m, 2.0)) - 0.5;
	} else {
		tmp = 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 <= 2.3:
		tmp = (0.225 * math.pow(x_m, 2.0)) - 0.5
	else:
		tmp = 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 <= 2.3)
		tmp = Float64(Float64(0.225 * (x_m ^ 2.0)) - 0.5);
	else
		tmp = Float64(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 <= 2.3)
		tmp = (0.225 * (x_m ^ 2.0)) - 0.5;
	else
		tmp = 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, 2.3], N[(N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(x$95$m / N[(x$95$m - N[Tan[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.2999999999999998

   1. Initial program 33.3%

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

   2. Taylor expanded in x around 0 70.1%

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

   if 2.2999999999999998 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.0%

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

4. Final simplification 78.0%

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

Alternative 6: 98.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.35:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.35) -0.5 (/ x_m (- x_m (tan x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.35) {
		tmp = -0.5;
	} else {
		tmp = 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 <= 1.35d0) then
        tmp = -0.5d0
    else
        tmp = 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 <= 1.35) {
		tmp = -0.5;
	} else {
		tmp = 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 <= 1.35:
		tmp = -0.5
	else:
		tmp = 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 <= 1.35)
		tmp = -0.5;
	else
		tmp = Float64(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 <= 1.35)
		tmp = -0.5;
	else
		tmp = 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, 1.35], -0.5, N[(x$95$m / N[(x$95$m - N[Tan[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.3500000000000001

   1. Initial program 33.3%

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

   2. Taylor expanded in x around 0 68.4%

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

   if 1.3500000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.35:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \]

Alternative 7: 98.5% accurate, 67.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.55:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.55) -0.5 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.55) {
		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 <= 1.55d0) 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 <= 1.55) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.55:
		tmp = -0.5
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.55)
		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 <= 1.55)
		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, 1.55], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 33.3%

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

   2. Taylor expanded in x around 0 68.4%

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

   if 1.55000000000000004 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.0%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.55:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 50.1% 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 51.5%

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

2. Taylor expanded in x around 0 50.2%

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

3. Final simplification 50.2%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023330 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))