sintan (problem 3.4.5)

Percentage Accurate: 50.9% → 100.0%

Time: 17.3s

Alternatives: 11

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

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x - \tan x\\ \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + 0.225 \cdot {x}^{2}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0} - \frac{\sin x}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- x (tan x))))
   (if (<= x 0.088)
     (-
      (+
       (* -0.009642857142857142 (pow x 4.0))
       (+ (* 0.00024107142857142857 (pow x 6.0)) (* 0.225 (pow x 2.0))))
      0.5)
     (- (/ x t_0) (/ (sin x) t_0)))))

C

x = abs(x);
double code(double x) {
	double t_0 = x - tan(x);
	double tmp;
	if (x <= 0.088) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + ((0.00024107142857142857 * pow(x, 6.0)) + (0.225 * pow(x, 2.0)))) - 0.5;
	} else {
		tmp = (x / t_0) - (sin(x) / t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(x)
    if (x <= 0.088d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + ((0.00024107142857142857d0 * (x ** 6.0d0)) + (0.225d0 * (x ** 2.0d0)))) - 0.5d0
    else
        tmp = (x / t_0) - (sin(x) / t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = x - Math.tan(x);
	double tmp;
	if (x <= 0.088) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + ((0.00024107142857142857 * Math.pow(x, 6.0)) + (0.225 * Math.pow(x, 2.0)))) - 0.5;
	} else {
		tmp = (x / t_0) - (Math.sin(x) / t_0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = x - math.tan(x)
	tmp = 0
	if x <= 0.088:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + ((0.00024107142857142857 * math.pow(x, 6.0)) + (0.225 * math.pow(x, 2.0)))) - 0.5
	else:
		tmp = (x / t_0) - (math.sin(x) / t_0)
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(x - tan(x))
	tmp = 0.0
	if (x <= 0.088)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(Float64(0.00024107142857142857 * (x ^ 6.0)) + Float64(0.225 * (x ^ 2.0)))) - 0.5);
	else
		tmp = Float64(Float64(x / t_0) - Float64(sin(x) / t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = x - tan(x);
	tmp = 0.0;
	if (x <= 0.088)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + ((0.00024107142857142857 * (x ^ 6.0)) + (0.225 * (x ^ 2.0)))) - 0.5;
	else
		tmp = (x / t_0) - (sin(x) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.088], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(0.225 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.087999999999999995

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0 64.6%

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

   if 0.087999999999999995 < 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 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + 0.225 \cdot {x}^{2}\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array} \]

Alternative 2: 100.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := x - \tan x\\ \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + \left(\left(1 + x \cdot \left(x \cdot 0.225\right)\right) + -1\right)\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t_0} - \frac{\sin x}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- x (tan x))))
   (if (<= x 0.088)
     (-
      (+
       (* -0.009642857142857142 (pow x 4.0))
       (+
        (* 0.00024107142857142857 (pow x 6.0))
        (+ (+ 1.0 (* x (* x 0.225))) -1.0)))
      0.5)
     (- (/ x t_0) (/ (sin x) t_0)))))

C

x = abs(x);
double code(double x) {
	double t_0 = x - tan(x);
	double tmp;
	if (x <= 0.088) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + ((0.00024107142857142857 * pow(x, 6.0)) + ((1.0 + (x * (x * 0.225))) + -1.0))) - 0.5;
	} else {
		tmp = (x / t_0) - (sin(x) / t_0);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x - tan(x)
    if (x <= 0.088d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + ((0.00024107142857142857d0 * (x ** 6.0d0)) + ((1.0d0 + (x * (x * 0.225d0))) + (-1.0d0)))) - 0.5d0
    else
        tmp = (x / t_0) - (sin(x) / t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = x - Math.tan(x);
	double tmp;
	if (x <= 0.088) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + ((0.00024107142857142857 * Math.pow(x, 6.0)) + ((1.0 + (x * (x * 0.225))) + -1.0))) - 0.5;
	} else {
		tmp = (x / t_0) - (Math.sin(x) / t_0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = x - math.tan(x)
	tmp = 0
	if x <= 0.088:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + ((0.00024107142857142857 * math.pow(x, 6.0)) + ((1.0 + (x * (x * 0.225))) + -1.0))) - 0.5
	else:
		tmp = (x / t_0) - (math.sin(x) / t_0)
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(x - tan(x))
	tmp = 0.0
	if (x <= 0.088)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(Float64(0.00024107142857142857 * (x ^ 6.0)) + Float64(Float64(1.0 + Float64(x * Float64(x * 0.225))) + -1.0))) - 0.5);
	else
		tmp = Float64(Float64(x / t_0) - Float64(sin(x) / t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = x - tan(x);
	tmp = 0.0;
	if (x <= 0.088)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + ((0.00024107142857142857 * (x ^ 6.0)) + ((1.0 + (x * (x * 0.225))) + -1.0))) - 0.5;
	else
		tmp = (x / t_0) - (sin(x) / t_0);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, 0.088], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[(x / t$95$0), $MachinePrecision] - N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.087999999999999995

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0 64.6%

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

      1. pow2 64.6%

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

      2. *-commutative 64.6%

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

      3. associate-*r* 64.6%

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

      4. expm1-log1p-u 64.6%

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

      5. expm1-def 64.6%

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

      6. log1p-udef 64.6%

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

      7. add-exp-log 64.6%

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

   4. Applied egg-rr 64.6%

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

   if 0.087999999999999995 < 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 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.088:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + \left(\left(1 + x \cdot \left(x \cdot 0.225\right)\right) + -1\right)\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x} - \frac{\sin x}{x - \tan x}\\ \end{array} \]

Alternative 3: 100.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.08:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + \left(\left(1 + x \cdot \left(x \cdot 0.225\right)\right) + -1\right)\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.08)
   (-
    (+
     (* -0.009642857142857142 (pow x 4.0))
     (+
      (* 0.00024107142857142857 (pow x 6.0))
      (+ (+ 1.0 (* x (* x 0.225))) -1.0)))
    0.5)
   (/ (- x (sin x)) (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.08) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + ((0.00024107142857142857 * pow(x, 6.0)) + ((1.0 + (x * (x * 0.225))) + -1.0))) - 0.5;
	} else {
		tmp = (x - sin(x)) / (x - tan(x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.08d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + ((0.00024107142857142857d0 * (x ** 6.0d0)) + ((1.0d0 + (x * (x * 0.225d0))) + (-1.0d0)))) - 0.5d0
    else
        tmp = (x - sin(x)) / (x - tan(x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.08) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + ((0.00024107142857142857 * Math.pow(x, 6.0)) + ((1.0 + (x * (x * 0.225))) + -1.0))) - 0.5;
	} else {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.08:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + ((0.00024107142857142857 * math.pow(x, 6.0)) + ((1.0 + (x * (x * 0.225))) + -1.0))) - 0.5
	else:
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.08)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(Float64(0.00024107142857142857 * (x ^ 6.0)) + Float64(Float64(1.0 + Float64(x * Float64(x * 0.225))) + -1.0))) - 0.5);
	else
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.08)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + ((0.00024107142857142857 * (x ^ 6.0)) + ((1.0 + (x * (x * 0.225))) + -1.0))) - 0.5;
	else
		tmp = (x - sin(x)) / (x - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.08], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(N[(0.00024107142857142857 * N[Power[x, 6.0], $MachinePrecision]), $MachinePrecision] + N[(N[(1.0 + N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0800000000000000017

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0 64.6%

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

      1. pow2 64.6%

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

      2. *-commutative 64.6%

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

      3. associate-*r* 64.6%

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

      4. expm1-log1p-u 64.6%

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

      5. expm1-def 64.6%

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

      6. log1p-udef 64.6%

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

      7. add-exp-log 64.6%

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

   4. Applied egg-rr 64.6%

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

   if 0.0800000000000000017 < x

   1. Initial program 100.0%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.08:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + \left(0.00024107142857142857 \cdot {x}^{6} + \left(\left(1 + x \cdot \left(x \cdot 0.225\right)\right) + -1\right)\right)\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 4: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\sin x - \tan x}{x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.6)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) (* x (* x 0.225))) -0.5)
   (- 1.0 (/ (- (sin x) (tan x)) x))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + (x * (x * 0.225))) + -0.5;
	} else {
		tmp = 1.0 - ((sin(x) - tan(x)) / x);
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.6d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (x * (x * 0.225d0))) + (-0.5d0)
    else
        tmp = 1.0d0 - ((sin(x) - tan(x)) / x)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.6) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + (x * (x * 0.225))) + -0.5;
	} else {
		tmp = 1.0 - ((Math.sin(x) - Math.tan(x)) / x);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.6:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + (x * (x * 0.225))) + -0.5
	else:
		tmp = 1.0 - ((math.sin(x) - math.tan(x)) / x)
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.6)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(x * Float64(x * 0.225))) + -0.5);
	else
		tmp = Float64(1.0 - Float64(Float64(sin(x) - tan(x)) / x));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.6)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + (x * (x * 0.225))) + -0.5;
	else
		tmp = 1.0 - ((sin(x) - tan(x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.6], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(1.0 - N[(N[(N[Sin[x], $MachinePrecision] - N[Tan[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0 64.1%

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

      1. sub-neg 64.1%

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

      2. fma-def 64.1%

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

      3. unpow2 64.1%

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

      4. metadata-eval 64.1%

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

   4. Simplified 64.1%

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

      1. fma-udef 64.1%

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

      2. +-commutative 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l* 64.1%

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

   6. Applied egg-rr 64.1%

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

   if 2.60000000000000009 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 99.7%

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

      1. associate--l+ 99.7%

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

      2. sub-neg 99.7%

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

      3. *-lft-identity 99.7%

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

      4. metadata-eval 99.7%

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

      5. cancel-sign-sub-inv 99.7%

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

      6. distribute-lft-out-- 99.7%

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

      7. mul-1-neg 99.7%

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

      8. remove-double-neg 99.7%

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

      9. *-commutative 99.7%

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

      10. associate-/r* 99.7%

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

      11. div-sub 99.7%

          \[\leadsto 1 + -1 \cdot \color{blue}{\frac{\sin x - \frac{\sin x}{\cos x}}{x}} \]

   4. Simplified 99.7%

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

      1. tan-quot 99.7%

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

      2. sub-neg 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. sub-neg 99.7%

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

   8. Simplified 99.7%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{\sin x - \tan x}{x}\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.03)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) (* x (* x 0.225))) -0.5)
   (/ (- x (sin x)) (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.03) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + (x * (x * 0.225))) + -0.5;
	} else {
		tmp = (x - sin(x)) / (x - tan(x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 0.03d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (x * (x * 0.225d0))) + (-0.5d0)
    else
        tmp = (x - sin(x)) / (x - tan(x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.03) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + (x * (x * 0.225))) + -0.5;
	} else {
		tmp = (x - Math.sin(x)) / (x - Math.tan(x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.03:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + (x * (x * 0.225))) + -0.5
	else:
		tmp = (x - math.sin(x)) / (x - math.tan(x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.03)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(x * Float64(x * 0.225))) + -0.5);
	else
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.03)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + (x * (x * 0.225))) + -0.5;
	else
		tmp = (x - sin(x)) / (x - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.03], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.029999999999999999

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0 64.1%

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

      1. sub-neg 64.1%

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

      2. fma-def 64.1%

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

      3. unpow2 64.1%

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

      4. metadata-eval 64.1%

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

   4. Simplified 64.1%

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

      1. fma-udef 64.1%

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

      2. +-commutative 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l* 64.1%

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

   6. Applied egg-rr 64.1%

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

   if 0.029999999999999999 < x

   1. Initial program 100.0%

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

4. Final simplification 75.0%

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

Alternative 6: 98.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.8:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{x - \tan x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.8)
   (+ (+ (* -0.009642857142857142 (pow x 4.0)) (* x (* x 0.225))) -0.5)
   (/ 1.0 (/ 1.0 (/ x (- x (tan x)))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.8) {
		tmp = ((-0.009642857142857142 * pow(x, 4.0)) + (x * (x * 0.225))) + -0.5;
	} else {
		tmp = 1.0 / (1.0 / (x / (x - tan(x))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.8d0) then
        tmp = (((-0.009642857142857142d0) * (x ** 4.0d0)) + (x * (x * 0.225d0))) + (-0.5d0)
    else
        tmp = 1.0d0 / (1.0d0 / (x / (x - tan(x))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.8) {
		tmp = ((-0.009642857142857142 * Math.pow(x, 4.0)) + (x * (x * 0.225))) + -0.5;
	} else {
		tmp = 1.0 / (1.0 / (x / (x - Math.tan(x))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.8:
		tmp = ((-0.009642857142857142 * math.pow(x, 4.0)) + (x * (x * 0.225))) + -0.5
	else:
		tmp = 1.0 / (1.0 / (x / (x - math.tan(x))))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.8)
		tmp = Float64(Float64(Float64(-0.009642857142857142 * (x ^ 4.0)) + Float64(x * Float64(x * 0.225))) + -0.5);
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(x / Float64(x - tan(x)))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.8)
		tmp = ((-0.009642857142857142 * (x ^ 4.0)) + (x * (x * 0.225))) + -0.5;
	else
		tmp = 1.0 / (1.0 / (x / (x - tan(x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.8], N[(N[(N[(-0.009642857142857142 * N[Power[x, 4.0], $MachinePrecision]), $MachinePrecision] + N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(1.0 / N[(1.0 / N[(x / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.7999999999999998

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0 64.1%

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

      1. sub-neg 64.1%

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

      2. fma-def 64.1%

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

      3. unpow2 64.1%

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

      4. metadata-eval 64.1%

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

   4. Simplified 64.1%

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

      1. fma-udef 64.1%

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

      2. +-commutative 64.1%

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

      3. *-commutative 64.1%

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

      4. associate-*l* 64.1%

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

   6. Applied egg-rr 64.1%

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

   if 2.7999999999999998 < 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. inv-pow 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 98.4%

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

      1. clear-num 98.4%

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

      2. inv-pow 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. unpow-1 98.4%

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

   10. Simplified 98.4%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.8:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + x \cdot \left(x \cdot 0.225\right)\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{x - \tan x}}}\\ \end{array} \]

Alternative 7: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{x - \tan x}}}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.25)
   (+ (* x (* x 0.225)) -0.5)
   (/ 1.0 (/ 1.0 (/ x (- x (tan x)))))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.25) {
		tmp = (x * (x * 0.225)) + -0.5;
	} else {
		tmp = 1.0 / (1.0 / (x / (x - tan(x))));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.25d0) then
        tmp = (x * (x * 0.225d0)) + (-0.5d0)
    else
        tmp = 1.0d0 / (1.0d0 / (x / (x - tan(x))))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.25) {
		tmp = (x * (x * 0.225)) + -0.5;
	} else {
		tmp = 1.0 / (1.0 / (x / (x - Math.tan(x))));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.25:
		tmp = (x * (x * 0.225)) + -0.5
	else:
		tmp = 1.0 / (1.0 / (x / (x - math.tan(x))))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(Float64(x * Float64(x * 0.225)) + -0.5);
	else
		tmp = Float64(1.0 / Float64(1.0 / Float64(x / Float64(x - tan(x)))));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.25)
		tmp = (x * (x * 0.225)) + -0.5;
	else
		tmp = 1.0 / (1.0 / (x / (x - tan(x))));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.25], N[(N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(1.0 / N[(1.0 / N[(x / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0 65.2%

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

      1. fma-neg 65.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, {x}^{2}, -0.5\right)} \]

      2. unpow2 65.2%

         \[\leadsto \mathsf{fma}\left(0.225, \color{blue}{x \cdot x}, -0.5\right) \]

      3. metadata-eval 65.2%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{-0.5}\right) \]

   4. Simplified 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
   5. Step-by-step derivation

      1. fma-udef 65.2%

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

      2. *-commutative 65.2%

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

      3. associate-*l* 65.2%

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

   6. Applied egg-rr 65.2%

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

   if 2.25 < 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. inv-pow 100.0%

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

   3. Applied egg-rr 100.0%

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

      1. unpow-1 100.0%

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

   5. Simplified 100.0%

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

   6. Taylor expanded in x around inf 98.4%

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

      1. clear-num 98.4%

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

      2. inv-pow 98.4%

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

   8. Applied egg-rr 98.4%

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

      1. unpow-1 98.4%

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

   10. Simplified 98.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{1}{\frac{x}{x - \tan x}}}\\ \end{array} \]

Alternative 8: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.25) (+ (* x (* x 0.225)) -0.5) (/ x (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.25) {
		tmp = (x * (x * 0.225)) + -0.5;
	} else {
		tmp = x / (x - tan(x));
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.25d0) then
        tmp = (x * (x * 0.225d0)) + (-0.5d0)
    else
        tmp = x / (x - tan(x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.25) {
		tmp = (x * (x * 0.225)) + -0.5;
	} else {
		tmp = x / (x - Math.tan(x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.25:
		tmp = (x * (x * 0.225)) + -0.5
	else:
		tmp = x / (x - math.tan(x))
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.25)
		tmp = Float64(Float64(x * Float64(x * 0.225)) + -0.5);
	else
		tmp = Float64(x / Float64(x - tan(x)));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.25)
		tmp = (x * (x * 0.225)) + -0.5;
	else
		tmp = x / (x - tan(x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.25], N[(N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(x / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.25

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0 65.2%

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

      1. fma-neg 65.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, {x}^{2}, -0.5\right)} \]

      2. unpow2 65.2%

         \[\leadsto \mathsf{fma}\left(0.225, \color{blue}{x \cdot x}, -0.5\right) \]

      3. metadata-eval 65.2%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{-0.5}\right) \]

   4. Simplified 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
   5. Step-by-step derivation

      1. fma-udef 65.2%

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

      2. *-commutative 65.2%

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

      3. associate-*l* 65.2%

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

   6. Applied egg-rr 65.2%

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

   if 2.25 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.4%

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

4. Final simplification 75.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.25:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x - \tan x}\\ \end{array} \]

Alternative 9: 98.8% accurate, 22.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 2.5) (+ (* x (* x 0.225)) -0.5) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = (x * (x * 0.225)) + -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.5d0) then
        tmp = (x * (x * 0.225d0)) + (-0.5d0)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 2.5) {
		tmp = (x * (x * 0.225)) + -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 2.5:
		tmp = (x * (x * 0.225)) + -0.5
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.5)
		tmp = Float64(Float64(x * Float64(x * 0.225)) + -0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.5)
		tmp = (x * (x * 0.225)) + -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.5], N[(N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.5

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0 65.2%

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

      1. fma-neg 65.2%

         \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, {x}^{2}, -0.5\right)} \]

      2. unpow2 65.2%

         \[\leadsto \mathsf{fma}\left(0.225, \color{blue}{x \cdot x}, -0.5\right) \]

      3. metadata-eval 65.2%

         \[\leadsto \mathsf{fma}\left(0.225, x \cdot x, \color{blue}{-0.5}\right) \]

   4. Simplified 65.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)} \]
   5. Step-by-step derivation

      1. fma-udef 65.2%

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

      2. *-commutative 65.2%

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

      3. associate-*l* 65.2%

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

   6. Applied egg-rr 65.2%

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

   if 2.5 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.4%

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

4. Final simplification 75.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.5:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 98.5% accurate, 67.6× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.55) -0.5 1.0))

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d0) then
        tmp = -0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.55) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.55:
		tmp = -0.5
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.55)
		tmp = -0.5;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.55)
		tmp = -0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.55], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 37.8%

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

   2. Taylor expanded in x around 0 63.3%

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

   if 1.55000000000000004 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 98.4%

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

4. Final simplification 74.0%

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

Alternative 11: 50.3% accurate, 207.0× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 -0.5)

C

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

Fortran

NOTE: x should be positive before calling this function
real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.5d0
end function

Java

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

Python

x = abs(x)
def code(x):
	return -0.5

Julia

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

MATLAB

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

Wolfram

NOTE: x should be positive before calling this function
code[x_] := -0.5

Derivation

1. Initial program 56.7%

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

2. Taylor expanded in x around 0 44.5%

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

3. Final simplification 44.5%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023293 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))