sintan (problem 3.4.5)

Percentage Accurate: 50.4% → 100.0%

Time: 13.5s

Alternatives: 7

Speedup: 207.0×

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: 50.4% 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

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.026:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x_m}^{4} + 0.225 \cdot {x_m}^{2}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \sin x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.026)
   (- (+ (* -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 <= 0.026) {
		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 <= 0.026d0) 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 <= 0.026) {
		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 <= 0.026:
		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 <= 0.026)
		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 <= 0.026)
		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, 0.026], 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 < 0.0259999999999999988

   1. Initial program 39.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 63.1%

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

   if 0.0259999999999999988 < x

   1. Initial program 99.9%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.026:\\ \;\;\;\;\left(-0.009642857142857142 \cdot {x}^{4} + 0.225 \cdot {x}^{2}\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.4:\\ \;\;\;\;\sqrt[3]{{x_m}^{6} \cdot 0.011390625} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\tan x_m - \sin x_m}{x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.4)
   (- (cbrt (* (pow x_m 6.0) 0.011390625)) 0.5)
   (+ 1.0 (/ (- (tan x_m) (sin x_m)) x_m))))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.4) {
		tmp = cbrt((pow(x_m, 6.0) * 0.011390625)) - 0.5;
	} else {
		tmp = 1.0 + ((tan(x_m) - sin(x_m)) / x_m);
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.4) {
		tmp = Math.cbrt((Math.pow(x_m, 6.0) * 0.011390625)) - 0.5;
	} else {
		tmp = 1.0 + ((Math.tan(x_m) - Math.sin(x_m)) / x_m);
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.4)
		tmp = Float64(cbrt(Float64((x_m ^ 6.0) * 0.011390625)) - 0.5);
	else
		tmp = Float64(1.0 + Float64(Float64(tan(x_m) - sin(x_m)) / x_m));
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.4], N[(N[Power[N[(N[Power[x$95$m, 6.0], $MachinePrecision] * 0.011390625), $MachinePrecision], 1/3], $MachinePrecision] - 0.5), $MachinePrecision], N[(1.0 + N[(N[(N[Tan[x$95$m], $MachinePrecision] - N[Sin[x$95$m], $MachinePrecision]), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 39.4%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 64.2%

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

      1. add-cbrt-cube 64.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.225 \cdot {x}^{2}\right) \cdot \left(0.225 \cdot {x}^{2}\right)\right) \cdot \left(0.225 \cdot {x}^{2}\right)}} - 0.5 \]

      2. pow1/3 64.1%

         \[\leadsto \color{blue}{{\left(\left(\left(0.225 \cdot {x}^{2}\right) \cdot \left(0.225 \cdot {x}^{2}\right)\right) \cdot \left(0.225 \cdot {x}^{2}\right)\right)}^{0.3333333333333333}} - 0.5 \]

      3. pow3 64.1%

         \[\leadsto {\color{blue}{\left({\left(0.225 \cdot {x}^{2}\right)}^{3}\right)}}^{0.3333333333333333} - 0.5 \]

      4. *-commutative 64.1%

         \[\leadsto {\left({\color{blue}{\left({x}^{2} \cdot 0.225\right)}}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      5. unpow-prod-down 64.1%

         \[\leadsto {\color{blue}{\left({\left({x}^{2}\right)}^{3} \cdot {0.225}^{3}\right)}}^{0.3333333333333333} - 0.5 \]

      6. pow3 64.1%

         \[\leadsto {\left(\color{blue}{\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot {x}^{2}\right)} \cdot {0.225}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      7. pow-sqr 64.1%

         \[\leadsto {\left(\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}} \cdot {x}^{2}\right) \cdot {0.225}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      8. metadata-eval 64.1%

         \[\leadsto {\left(\left({x}^{\color{blue}{4}} \cdot {x}^{2}\right) \cdot {0.225}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      9. pow-prod-up 64.1%

         \[\leadsto {\left(\color{blue}{{x}^{\left(4 + 2\right)}} \cdot {0.225}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      10. metadata-eval 64.1%

          \[\leadsto {\left({x}^{\color{blue}{6}} \cdot {0.225}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      11. metadata-eval 64.1%

          \[\leadsto {\left({x}^{6} \cdot \color{blue}{0.011390625}\right)}^{0.3333333333333333} - 0.5 \]

   5. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{{\left({x}^{6} \cdot 0.011390625\right)}^{0.3333333333333333}} - 0.5 \]
   6. Step-by-step derivation

      1. unpow1/3 64.1%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{6} \cdot 0.011390625}} - 0.5 \]

   7. Simplified 64.1%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{6} \cdot 0.011390625}} - 0.5 \]

   if 2.39999999999999991 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 97.9%

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

      1. associate--l+ 97.9%

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

      2. sub-neg 97.9%

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

      3. *-lft-identity 97.9%

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

      4. metadata-eval 97.9%

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

      5. cancel-sign-sub-inv 97.9%

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

      6. distribute-lft-out-- 97.9%

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

      7. mul-1-neg 97.9%

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

      8. remove-double-neg 97.9%

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

      9. associate-/l/ 97.9%

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

      10. div-sub 97.9%

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

      11. mul-1-neg 97.9%

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

   5. Simplified 97.9%

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

      1. tan-quot 97.9%

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

      2. sub-neg 97.9%

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

   7. Applied egg-rr 97.9%

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

      1. sub-neg 97.9%

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

   9. Simplified 97.9%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;\sqrt[3]{{x}^{6} \cdot 0.011390625} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 0.0052:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m - \sin x_m}{x_m - \tan x_m}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 0.0052)
   (- (* 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 <= 0.0052) {
		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 <= 0.0052d0) 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 <= 0.0052) {
		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 <= 0.0052:
		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 <= 0.0052)
		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 <= 0.0052)
		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, 0.0052], 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 < 0.0051999999999999998

   1. Initial program 39.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 64.4%

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

   if 0.0051999999999999998 < x

   1. Initial program 99.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0052:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 98.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.6:\\ \;\;\;\;\sqrt[3]{{x_m}^{6} \cdot 0.011390625} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.6) (- (cbrt (* (pow x_m 6.0) 0.011390625)) 0.5) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.6) {
		tmp = cbrt((pow(x_m, 6.0) * 0.011390625)) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.6) {
		tmp = Math.cbrt((Math.pow(x_m, 6.0) * 0.011390625)) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.6)
		tmp = Float64(cbrt(Float64((x_m ^ 6.0) * 0.011390625)) - 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.6], N[(N[Power[N[(N[Power[x$95$m, 6.0], $MachinePrecision] * 0.011390625), $MachinePrecision], 1/3], $MachinePrecision] - 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 39.4%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 64.2%

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

      1. add-cbrt-cube 64.1%

         \[\leadsto \color{blue}{\sqrt[3]{\left(\left(0.225 \cdot {x}^{2}\right) \cdot \left(0.225 \cdot {x}^{2}\right)\right) \cdot \left(0.225 \cdot {x}^{2}\right)}} - 0.5 \]

      2. pow1/3 64.1%

         \[\leadsto \color{blue}{{\left(\left(\left(0.225 \cdot {x}^{2}\right) \cdot \left(0.225 \cdot {x}^{2}\right)\right) \cdot \left(0.225 \cdot {x}^{2}\right)\right)}^{0.3333333333333333}} - 0.5 \]

      3. pow3 64.1%

         \[\leadsto {\color{blue}{\left({\left(0.225 \cdot {x}^{2}\right)}^{3}\right)}}^{0.3333333333333333} - 0.5 \]

      4. *-commutative 64.1%

         \[\leadsto {\left({\color{blue}{\left({x}^{2} \cdot 0.225\right)}}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      5. unpow-prod-down 64.1%

         \[\leadsto {\color{blue}{\left({\left({x}^{2}\right)}^{3} \cdot {0.225}^{3}\right)}}^{0.3333333333333333} - 0.5 \]

      6. pow3 64.1%

         \[\leadsto {\left(\color{blue}{\left(\left({x}^{2} \cdot {x}^{2}\right) \cdot {x}^{2}\right)} \cdot {0.225}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      7. pow-sqr 64.1%

         \[\leadsto {\left(\left(\color{blue}{{x}^{\left(2 \cdot 2\right)}} \cdot {x}^{2}\right) \cdot {0.225}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      8. metadata-eval 64.1%

         \[\leadsto {\left(\left({x}^{\color{blue}{4}} \cdot {x}^{2}\right) \cdot {0.225}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      9. pow-prod-up 64.1%

         \[\leadsto {\left(\color{blue}{{x}^{\left(4 + 2\right)}} \cdot {0.225}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      10. metadata-eval 64.1%

          \[\leadsto {\left({x}^{\color{blue}{6}} \cdot {0.225}^{3}\right)}^{0.3333333333333333} - 0.5 \]

      11. metadata-eval 64.1%

          \[\leadsto {\left({x}^{6} \cdot \color{blue}{0.011390625}\right)}^{0.3333333333333333} - 0.5 \]

   5. Applied egg-rr 64.1%

      \[\leadsto \color{blue}{{\left({x}^{6} \cdot 0.011390625\right)}^{0.3333333333333333}} - 0.5 \]
   6. Step-by-step derivation

      1. unpow1/3 64.1%

         \[\leadsto \color{blue}{\sqrt[3]{{x}^{6} \cdot 0.011390625}} - 0.5 \]

   7. Simplified 64.1%

      \[\leadsto \color{blue}{\sqrt[3]{{x}^{6} \cdot 0.011390625}} - 0.5 \]

   if 2.60000000000000009 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 97.2%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;\sqrt[3]{{x}^{6} \cdot 0.011390625} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ \begin{array}{l} \mathbf{if}\;x_m \leq 2.6:\\ \;\;\;\;0.225 \cdot {x_m}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
(FPCore (x_m)
 :precision binary64
 (if (<= x_m 2.6) (- (* 0.225 (pow x_m 2.0)) 0.5) 1.0))

C

x_m = fabs(x);
double code(double x_m) {
	double tmp;
	if (x_m <= 2.6) {
		tmp = (0.225 * pow(x_m, 2.0)) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

x_m = abs(x)
real(8) function code(x_m)
    real(8), intent (in) :: x_m
    real(8) :: tmp
    if (x_m <= 2.6d0) then
        tmp = (0.225d0 * (x_m ** 2.0d0)) - 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

x_m = Math.abs(x);
public static double code(double x_m) {
	double tmp;
	if (x_m <= 2.6) {
		tmp = (0.225 * Math.pow(x_m, 2.0)) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x_m = math.fabs(x)
def code(x_m):
	tmp = 0
	if x_m <= 2.6:
		tmp = (0.225 * math.pow(x_m, 2.0)) - 0.5
	else:
		tmp = 1.0
	return tmp

Julia

x_m = abs(x)
function code(x_m)
	tmp = 0.0
	if (x_m <= 2.6)
		tmp = Float64(Float64(0.225 * (x_m ^ 2.0)) - 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x_m = abs(x);
function tmp_2 = code(x_m)
	tmp = 0.0;
	if (x_m <= 2.6)
		tmp = (0.225 * (x_m ^ 2.0)) - 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
code[x$95$m_] := If[LessEqual[x$95$m, 2.6], N[(N[(0.225 * N[Power[x$95$m, 2.0], $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.60000000000000009

   1. Initial program 39.4%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 64.2%

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

   if 2.60000000000000009 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 97.2%

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

4. Final simplification 73.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.6:\\ \;\;\;\;0.225 \cdot {x}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 98.5% accurate, 34.4× 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 39.4%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0 62.3%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

      \[\frac{x - \sin x}{x - \tan x} \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf 97.2%

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

4. Final simplification 71.7%

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

Alternative 7: 50.7% 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 55.7%

   \[\frac{x - \sin x}{x - \tan x} \]
2. Add Preprocessing

3. Taylor expanded in x around 0 46.0%

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

4. Final simplification 46.0%

   \[\leadsto -0.5 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024018 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))