sintan (problem 3.4.5)

Percentage Accurate: 50.9% → 100.0%

Time: 13.5s

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: 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, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.027:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x_m}^{4} + 0.225 \cdot {x_m}^{2}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \sin x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.027)
   (- (+ (* -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 <= 0.027) {
		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 <= 0.027d0) 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 <= 0.027) {
		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 <= 0.027:
		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 <= 0.027)
		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 <= 0.027)
		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, 0.027], 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 < 0.0269999999999999997

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 62.9%

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

   if 0.0269999999999999997 < x

   1. Initial program 100.0%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.027:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot {x}^{2}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0044:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \sin x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0044)
   (- (* 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 <= 0.0044) {
		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 <= 0.0044d0) 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 <= 0.0044) {
		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 <= 0.0044:
		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 <= 0.0044)
		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 <= 0.0044)
		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, 0.0044], 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 < 0.00440000000000000027

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 64.2%

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

   if 0.00440000000000000027 < x

   1. Initial program 100.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.45:\\ \;\;\;\;{\left(x_m \cdot \sqrt{0.225}\right)}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{\tan x_m}{x_m}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 1.45)
   (- (pow (* x_m (sqrt 0.225)) 2.0) 0.5)
   (/ 1.0 (- 1.0 (/ (tan x_m) x_m)))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.45) {
		tmp = pow((x_m * sqrt(0.225)), 2.0) - 0.5;
	} else {
		tmp = 1.0 / (1.0 - (tan(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 <= 1.45d0) then
        tmp = ((x_m * sqrt(0.225d0)) ** 2.0d0) - 0.5d0
    else
        tmp = 1.0d0 / (1.0d0 - (tan(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 <= 1.45) {
		tmp = Math.pow((x_m * Math.sqrt(0.225)), 2.0) - 0.5;
	} else {
		tmp = 1.0 / (1.0 - (Math.tan(x_m) / x_m));
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.45:
		tmp = math.pow((x_m * math.sqrt(0.225)), 2.0) - 0.5
	else:
		tmp = 1.0 / (1.0 - (math.tan(x_m) / x_m))
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.45)
		tmp = Float64((Float64(x_m * sqrt(0.225)) ^ 2.0) - 0.5);
	else
		tmp = Float64(1.0 / Float64(1.0 - Float64(tan(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 <= 1.45)
		tmp = ((x_m * sqrt(0.225)) ^ 2.0) - 0.5;
	else
		tmp = 1.0 / (1.0 - (tan(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, 1.45], N[(N[Power[N[(x$95$m * N[Sqrt[0.225], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - 0.5), $MachinePrecision], N[(1.0 / N[(1.0 - N[(N[Tan[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 64.2%

      \[\leadsto \color{blue}{0.225 \cdot {x}^{2} - 0.5} \]
   3. Step-by-step derivation

      1. add-sqr-sqrt 64.2%

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

      2. pow2 64.2%

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

      3. *-commutative 64.2%

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

      4. sqrt-prod 64.2%

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

      5. unpow2 64.2%

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

      6. sqrt-prod 32.8%

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

      7. add-sqr-sqrt 64.2%

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

   4. Applied egg-rr 64.2%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 97.6%

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

      1. clear-num 97.6%

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

      2. associate-/r/ 97.4%

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

   4. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{\frac{1}{x - \tan x} \cdot x} \]
   5. Step-by-step derivation

      1. associate-/r/ 97.6%

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

      2. div-sub 97.6%

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

      3. *-inverses 97.6%

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

   6. Applied egg-rr 97.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;{\left(x \cdot \sqrt{0.225}\right)}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{\tan x}{x}}\\ \end{array} \]

Alternative 4: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.45:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{\tan x_m}{x_m}}\\ \end{array} \end{array} \]

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)
   (/ 1.0 (- 1.0 (/ (tan x_m) 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 = 1.0 / (1.0 - (tan(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 <= 1.45d0) then
        tmp = (0.225d0 * (x_m ** 2.0d0)) - 0.5d0
    else
        tmp = 1.0d0 / (1.0d0 - (tan(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 <= 1.45) {
		tmp = (0.225 * Math.pow(x_m, 2.0)) - 0.5;
	} else {
		tmp = 1.0 / (1.0 - (Math.tan(x_m) / 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 = 1.0 / (1.0 - (math.tan(x_m) / 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(1.0 / Float64(1.0 - Float64(tan(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 <= 1.45)
		tmp = (0.225 * (x_m ^ 2.0)) - 0.5;
	else
		tmp = 1.0 / (1.0 - (tan(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, 1.45], N[(N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(1.0 / N[(1.0 - N[(N[Tan[x$95$m], $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.44999999999999996

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 64.2%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 97.6%

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

      1. clear-num 97.6%

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

      2. associate-/r/ 97.4%

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

   4. Applied egg-rr 97.4%

      \[\leadsto \color{blue}{\frac{1}{x - \tan x} \cdot x} \]
   5. Step-by-step derivation

      1. associate-/r/ 97.6%

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

      2. div-sub 97.6%

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

      3. *-inverses 97.6%

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

   6. Applied egg-rr 97.6%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.45:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{1 - \frac{\tan x}{x}}\\ \end{array} \]

Alternative 5: 98.8% accurate, 1.9× speedup

Math

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

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 38.0%

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

   2. Taylor expanded in x around 0 64.2%

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

   if 1.44999999999999996 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 97.6%

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

4. Final simplification 72.2%

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

Alternative 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 38.0%

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

   2. Taylor expanded in x around 0 63.0%

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

   if 1.3500000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 97.6%

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

4. Final simplification 71.2%

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

Alternative 7: 98.5% accurate, 67.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m) :precision binary64 (if (<= x_m 1.6) -0.5 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 1.6) {
		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 <= 1.6d0) 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 <= 1.6) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 1.6:
		tmp = -0.5
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 1.6)
		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 <= 1.6)
		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, 1.6], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 38.0%

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

   2. Taylor expanded in x around 0 63.0%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 97.6%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 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 52.7%

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

2. Taylor expanded in x around 0 48.3%

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

3. Final simplification 48.3%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023326 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))