sintan (problem 3.4.5)

Percentage Accurate: 50.9% → 100.0%

Time: 12.6s

Alternatives: 7

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: 50.9% 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, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.095:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x_m}^{4} + \left(\left(e^{\mathsf{log1p}\left(0.00024107142857142857 \cdot {x_m}^{6}\right)} + -1\right) + 0.225 \cdot {x_m}^{2}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x_m - \tan x_m}{x_m - \sin x_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.095)
   (-
    (+
     (* -0.009642857142857142 (pow x_m 4.0))
     (+
      (+ (exp (log1p (* 0.00024107142857142857 (pow x_m 6.0)))) -1.0)
      (* 0.225 (pow x_m 2.0))))
    0.5)
   (/ 1.0 (/ (- x_m (tan x_m)) (- x_m (sin x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.095) {
		tmp = ((-0.009642857142857142 * pow(x_m, 4.0)) + ((exp(log1p((0.00024107142857142857 * pow(x_m, 6.0)))) + -1.0) + (0.225 * pow(x_m, 2.0)))) - 0.5;
	} else {
		tmp = 1.0 / ((x_m - tan(x_m)) / (x_m - sin(x_m)));
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 0.095) {
		tmp = ((-0.009642857142857142 * Math.pow(x_m, 4.0)) + ((Math.exp(Math.log1p((0.00024107142857142857 * Math.pow(x_m, 6.0)))) + -1.0) + (0.225 * Math.pow(x_m, 2.0)))) - 0.5;
	} else {
		tmp = 1.0 / ((x_m - Math.tan(x_m)) / (x_m - Math.sin(x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.095:
		tmp = ((-0.009642857142857142 * math.pow(x_m, 4.0)) + ((math.exp(math.log1p((0.00024107142857142857 * math.pow(x_m, 6.0)))) + -1.0) + (0.225 * math.pow(x_m, 2.0)))) - 0.5
	else:
		tmp = 1.0 / ((x_m - math.tan(x_m)) / (x_m - math.sin(x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.095)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x_m ^ 4.0)) + Float64(Float64(exp(log1p(Float64(0.00024107142857142857 * (x_m ^ 6.0)))) + -1.0) + Float64(0.225 * (x_m ^ 2.0)))) - 0.5);
	else
		tmp = Float64(1.0 / Float64(Float64(x_m - tan(x_m)) / Float64(x_m - sin(x_m))));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.095], N[(N[(N[(-0.009642857142857142 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(N[Exp[N[Log[1 + N[(0.00024107142857142857 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision] + -1.0), $MachinePrecision] + N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(1.0 / N[(N[(x$95$m - N[Tan[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m - N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 33.2%

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

   2. Taylor expanded in x around 0 69.2%

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

      1. expm1-log1p-u 69.2%

         \[\leadsto \left(-0.009642857142857142 \cdot {x}^{4} + \left(\color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(0.00024107142857142857 \cdot {x}^{6}\right)\right)} + 0.225 \cdot {x}^{2}\right)\right) - 0.5 \]

      2. expm1-udef 69.2%

         \[\leadsto \left(-0.009642857142857142 \cdot {x}^{4} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.00024107142857142857 \cdot {x}^{6}\right)} - 1\right)} + 0.225 \cdot {x}^{2}\right)\right) - 0.5 \]

   4. Applied egg-rr 69.2%

      \[\leadsto \left(-0.009642857142857142 \cdot {x}^{4} + \left(\color{blue}{\left(e^{\mathsf{log1p}\left(0.00024107142857142857 \cdot {x}^{6}\right)} - 1\right)} + 0.225 \cdot {x}^{2}\right)\right) - 0.5 \]

   if 0.095000000000000001 < x

   1. Initial program 100.0%

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

      1. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \]

      2. associate-/r/ 99.8%

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

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{x - \tan x} \cdot \left(x - \sin x\right)} \]
   4. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(\left(e^{\mathsf{log1p}\left(0.00024107142857142857 \cdot {x}^{6}\right)} + -1\right) + 0.225 \cdot {x}^{2}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.095:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x_m}^{4} + \left(0.00024107142857142857 \cdot {x_m}^{6} + 0.225 \cdot {x_m}^{2}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x_m - \tan x_m}{x_m - \sin x_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.095)
   (-
    (+
     (* -0.009642857142857142 (pow x_m 4.0))
     (+ (* 0.00024107142857142857 (pow x_m 6.0)) (* 0.225 (pow x_m 2.0))))
    0.5)
   (/ 1.0 (/ (- x_m (tan x_m)) (- x_m (sin x_m))))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 0.095) {
		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 = 1.0 / ((x_m - tan(x_m)) / (x_m - sin(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 <= 0.095d0) then
        tmp = (((-0.009642857142857142d0) * (x_m ** 4.0d0)) + ((0.00024107142857142857d0 * (x_m ** 6.0d0)) + (0.225d0 * (x_m ** 2.0d0)))) - 0.5d0
    else
        tmp = 1.0d0 / ((x_m - tan(x_m)) / (x_m - sin(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 <= 0.095) {
		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 = 1.0 / ((x_m - Math.tan(x_m)) / (x_m - Math.sin(x_m)));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 0.095:
		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 = 1.0 / ((x_m - math.tan(x_m)) / (x_m - math.sin(x_m)))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 0.095)
		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(1.0 / Float64(Float64(x_m - tan(x_m)) / Float64(x_m - sin(x_m))));
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 0.095)
		tmp = ((-0.009642857142857142 * (x_m ^ 4.0)) + ((0.00024107142857142857 * (x_m ^ 6.0)) + (0.225 * (x_m ^ 2.0)))) - 0.5;
	else
		tmp = 1.0 / ((x_m - tan(x_m)) / (x_m - sin(x_m)));
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 0.095], 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[(1.0 / N[(N[(x$95$m - N[Tan[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m - N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.095000000000000001

   1. Initial program 33.2%

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

   2. Taylor expanded in x around 0 69.2%

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

   if 0.095000000000000001 < x

   1. Initial program 100.0%

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

      1. clear-num 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \]

      2. associate-/r/ 99.8%

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

   3. Applied egg-rr 99.8%

      \[\leadsto \color{blue}{\frac{1}{x - \tan x} \cdot \left(x - \sin x\right)} \]
   4. Step-by-step derivation

      1. associate-/r/ 100.0%

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 76.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.095:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + 0.225 \cdot {x}^{2}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.005:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x_m - \tan x_m}{x_m - \sin x_m}}\\ \end{array} \end{array} \]

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)
   (/ 1.0 (/ (- x_m (tan x_m)) (- x_m (sin 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 = 1.0 / ((x_m - tan(x_m)) / (x_m - sin(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 = 1.0d0 / ((x_m - tan(x_m)) / (x_m - sin(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 = 1.0 / ((x_m - Math.tan(x_m)) / (x_m - Math.sin(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 = 1.0 / ((x_m - math.tan(x_m)) / (x_m - math.sin(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(1.0 / Float64(Float64(x_m - tan(x_m)) / Float64(x_m - sin(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 = 1.0 / ((x_m - tan(x_m)) / (x_m - sin(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[(1.0 / N[(N[(x$95$m - N[Tan[x$95$m], $MachinePrecision]), $MachinePrecision] / N[(x$95$m - N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0050000000000000001

   1. Initial program 32.9%

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

   2. Taylor expanded in x around 0 70.0%

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

   if 0.0050000000000000001 < x

   1. Initial program 99.8%

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

      1. clear-num 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \]

      2. associate-/r/ 99.6%

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

   3. Applied egg-rr 99.6%

      \[\leadsto \color{blue}{\frac{1}{x - \tan x} \cdot \left(x - \sin x\right)} \]
   4. Step-by-step derivation

      1. associate-/r/ 99.9%

         \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{x - \tan x}{x - \sin x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.005:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{x - \tan x}{x - \sin x}}\\ \end{array} \]

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 32.9%

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

   2. Taylor expanded in x around 0 70.0%

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

   if 0.0050000000000000001 < x

   1. Initial program 99.8%

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

4. Final simplification 77.1%

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

Alternative 5: 98.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 7.8:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 7.8) (- (* 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 <= 7.8) {
		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 <= 7.8d0) 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 <= 7.8) {
		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 <= 7.8:
		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 <= 7.8)
		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 <= 7.8)
		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, 7.8], 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 < 7.79999999999999982

   1. Initial program 33.2%

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

   2. Taylor expanded in x around 0 69.9%

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

   if 7.79999999999999982 < 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 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 7.8:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 98.4% 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 33.2%

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

   2. Taylor expanded in x around 0 68.2%

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

   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 75.4%

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

Alternative 7: 50.2% 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 48.8%

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

2. Taylor expanded in x around 0 52.6%

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

3. Final simplification 52.6%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023331 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))