sintan (problem 3.4.5)

Percentage Accurate: 50.9% → 100.0%

Time: 13.7s

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, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.085:\\ \;\;\;\;\left(0.225 \cdot {x_m}^{2} + 0.00024107142857142857 \cdot {x_m}^{6}\right) + \left(-0.009642857142857142 \cdot {x_m}^{4} - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \sin x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.085)
   (+
    (+ (* 0.225 (pow x_m 2.0)) (* 0.00024107142857142857 (pow x_m 6.0)))
    (- (* -0.009642857142857142 (pow x_m 4.0)) 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 <= 0.085) {
		tmp = ((0.225 * pow(x_m, 2.0)) + (0.00024107142857142857 * pow(x_m, 6.0))) + ((-0.009642857142857142 * pow(x_m, 4.0)) - 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 <= 0.085d0) then
        tmp = ((0.225d0 * (x_m ** 2.0d0)) + (0.00024107142857142857d0 * (x_m ** 6.0d0))) + (((-0.009642857142857142d0) * (x_m ** 4.0d0)) - 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 <= 0.085) {
		tmp = ((0.225 * Math.pow(x_m, 2.0)) + (0.00024107142857142857 * Math.pow(x_m, 6.0))) + ((-0.009642857142857142 * Math.pow(x_m, 4.0)) - 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 <= 0.085:
		tmp = ((0.225 * math.pow(x_m, 2.0)) + (0.00024107142857142857 * math.pow(x_m, 6.0))) + ((-0.009642857142857142 * math.pow(x_m, 4.0)) - 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 <= 0.085)
		tmp = Float64(Float64(Float64(0.225 * (x_m ^ 2.0)) + Float64(0.00024107142857142857 * (x_m ^ 6.0))) + Float64(Float64(-0.009642857142857142 * (x_m ^ 4.0)) - 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 <= 0.085)
		tmp = ((0.225 * (x_m ^ 2.0)) + (0.00024107142857142857 * (x_m ^ 6.0))) + ((-0.009642857142857142 * (x_m ^ 4.0)) - 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, 0.085], N[(N[(N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(0.00024107142857142857 * N[Power[x$95$m, 6.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-0.009642857142857142 * N[Power[x$95$m, 4.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision]), $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 < 0.0850000000000000061

   1. Initial program 29.4%

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

   2. Taylor expanded in x around 0 72.1%

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

      1. +-commutative 72.1%

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

      2. associate--l+ 72.1%

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

      3. +-commutative 72.1%

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

      4. fma-def 72.1%

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

      5. fma-neg 72.1%

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

      6. metadata-eval 72.1%

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

   4. Simplified 72.1%

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

   5. Taylor expanded in x around 0 72.1%

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

      1. fma-udef 72.1%

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

   7. Applied egg-rr 72.1%

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

   if 0.0850000000000000061 < x

   1. Initial program 100.0%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.085:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + 0.00024107142857142857 \cdot {x}^{6}\right) + \left(-0.009642857142857142 \cdot {x}^{4} - 0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 2: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.03:\\ \;\;\;\;\left(0.225 \cdot {x_m}^{2} + -0.009642857142857142 \cdot {x_m}^{4}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \sin x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.03)
   (- (+ (* 0.225 (pow x_m 2.0)) (* -0.009642857142857142 (pow x_m 4.0))) 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 <= 0.03) {
		tmp = ((0.225 * pow(x_m, 2.0)) + (-0.009642857142857142 * pow(x_m, 4.0))) - 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 <= 0.03d0) then
        tmp = ((0.225d0 * (x_m ** 2.0d0)) + ((-0.009642857142857142d0) * (x_m ** 4.0d0))) - 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 <= 0.03) {
		tmp = ((0.225 * Math.pow(x_m, 2.0)) + (-0.009642857142857142 * Math.pow(x_m, 4.0))) - 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 <= 0.03:
		tmp = ((0.225 * math.pow(x_m, 2.0)) + (-0.009642857142857142 * math.pow(x_m, 4.0))) - 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 <= 0.03)
		tmp = Float64(Float64(Float64(0.225 * (x_m ^ 2.0)) + Float64(-0.009642857142857142 * (x_m ^ 4.0))) - 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 <= 0.03)
		tmp = ((0.225 * (x_m ^ 2.0)) + (-0.009642857142857142 * (x_m ^ 4.0))) - 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, 0.03], N[(N[(N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] + N[(-0.009642857142857142 * N[Power[x$95$m, 4.0], $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 < 0.029999999999999999

   1. Initial program 29.1%

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

   2. Taylor expanded in x around 0 71.9%

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

   if 0.029999999999999999 < x

   1. Initial program 99.8%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.03:\\ \;\;\;\;\left(0.225 \cdot {x}^{2} + -0.009642857142857142 \cdot {x}^{4}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0037:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \sin x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0037)
   (- (* 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 <= 0.0037) {
		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 <= 0.0037d0) 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 <= 0.0037) {
		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 <= 0.0037:
		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 <= 0.0037)
		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 <= 0.0037)
		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, 0.0037], 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 < 0.0037000000000000002

   1. Initial program 29.1%

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

   2. Taylor expanded in x around 0 72.8%

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

   if 0.0037000000000000002 < x

   1. Initial program 99.8%

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

4. Final simplification 79.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0037:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 4: 98.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.45:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.45) (- (* 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 <= 1.45) {
		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 <= 1.45d0) 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 <= 1.45) {
		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 <= 1.45:
		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 <= 1.45)
		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 <= 1.45)
		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, 1.45], 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 < 1.44999999999999996

   1. Initial program 29.4%

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

   2. Taylor expanded in x around 0 72.7%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 95.7%

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

4. Final simplification 78.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \]

Alternative 5: 98.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.32:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.32) -0.5 (/ x_m (- x_m (tan x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.32) {
		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 <= 1.32d0) 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 <= 1.32) {
		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 <= 1.32:
		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 <= 1.32)
		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 <= 1.32)
		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, 1.32], -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 < 1.32000000000000006

   1. Initial program 29.4%

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

   2. Taylor expanded in x around 0 71.5%

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

   if 1.32000000000000006 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 95.7%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.32:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \]

Alternative 6: 98.3% 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 29.4%

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

   2. Taylor expanded in x around 0 71.5%

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

   if 1.55000000000000004 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 95.6%

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

4. Final simplification 77.2%

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

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 46.0%

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

2. Taylor expanded in x around 0 55.1%

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

3. Final simplification 55.1%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023335 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))