sintan (problem 3.4.5)

Percentage Accurate: 50.0% → 99.9%

Time: 18.2s

Alternatives: 6

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: 50.0% 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: 99.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x - x}{\mathsf{fma}\left(\sin x, \frac{1}{\cos x}, -x\right)}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0044)
   (fma 0.225 (* x x) -0.5)
   (/ (- (sin x) x) (fma (sin x) (/ 1.0 (cos x)) (- x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = fma(0.225, (x * x), -0.5);
	} else {
		tmp = (sin(x) - x) / fma(sin(x), (1.0 / cos(x)), -x);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0044)
		tmp = fma(0.225, Float64(x * x), -0.5);
	else
		tmp = Float64(Float64(sin(x) - x) / fma(sin(x), Float64(1.0 / cos(x)), Float64(-x)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0044], N[(0.225 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision] / N[(N[Sin[x], $MachinePrecision] * N[(1.0 / N[Cos[x], $MachinePrecision]), $MachinePrecision] + (-x)), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00440000000000000027

   1. Initial program 34.4%

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

      1. sub-neg 34.4%

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

      2. +-commutative 34.4%

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

      3. neg-sub0 34.4%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 34.4%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 34.4%

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

      6. neg-mul-1 34.4%

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

      7. sub-neg 34.4%

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

      8. +-commutative 34.4%

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

      9. neg-sub0 34.4%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 34.4%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 34.4%

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

      12. neg-mul-1 34.4%

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

      13. times-frac 34.4%

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

      14. metadata-eval 34.4%

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

      15. *-lft-identity 34.4%

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

   3. Simplified 34.4%

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

   4. Taylor expanded in x around 0 67.3%

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

      1. fma-neg 67.3%

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

      2. unpow2 67.3%

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

      3. metadata-eval 67.3%

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

   6. Simplified 67.3%

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

   if 0.00440000000000000027 < x

   1. Initial program 99.9%

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

      1. sub-neg 99.9%

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

      2. +-commutative 99.9%

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

      3. neg-sub0 99.9%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 99.9%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 99.9%

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

      6. neg-mul-1 99.9%

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

      7. sub-neg 99.9%

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

      8. +-commutative 99.9%

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

      9. neg-sub0 99.9%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 99.9%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 99.9%

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

      12. neg-mul-1 99.9%

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

      13. times-frac 99.9%

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

      14. metadata-eval 99.9%

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

      15. *-lft-identity 99.9%

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

   3. Simplified 99.9%

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

      1. tan-quot 99.9%

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

      2. div-inv 99.9%

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

      3. fma-neg 99.9%

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

   5. Applied egg-rr 99.9%

      \[\leadsto \frac{\sin x - x}{\color{blue}{\mathsf{fma}\left(\sin x, \frac{1}{\cos x}, -x\right)}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0044:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x - x}{\mathsf{fma}\left(\sin x, \frac{1}{\cos x}, -x\right)}\\ \end{array} \]

Alternative 2: 99.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 0.0048:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 0.0048) (fma 0.225 (* x x) -0.5) (/ (- x (sin x)) (- x (tan x)))))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0048) {
		tmp = fma(0.225, (x * x), -0.5);
	} else {
		tmp = (x - sin(x)) / (x - tan(x));
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 0.0048)
		tmp = fma(0.225, Float64(x * x), -0.5);
	else
		tmp = Float64(Float64(x - sin(x)) / Float64(x - tan(x)));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 0.0048], N[(0.225 * N[(x * x), $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 < 0.00479999999999999958

   1. Initial program 34.4%

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

      1. sub-neg 34.4%

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

      2. +-commutative 34.4%

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

      3. neg-sub0 34.4%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 34.4%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 34.4%

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

      6. neg-mul-1 34.4%

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

      7. sub-neg 34.4%

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

      8. +-commutative 34.4%

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

      9. neg-sub0 34.4%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 34.4%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 34.4%

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

      12. neg-mul-1 34.4%

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

      13. times-frac 34.4%

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

      14. metadata-eval 34.4%

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

      15. *-lft-identity 34.4%

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

   3. Simplified 34.4%

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

   4. Taylor expanded in x around 0 67.3%

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

      1. fma-neg 67.3%

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

      2. unpow2 67.3%

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

      3. metadata-eval 67.3%

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

   6. Simplified 67.3%

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

   if 0.00479999999999999958 < x

   1. Initial program 99.9%

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

4. Final simplification 74.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0048:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \sin x}{x - \tan x}\\ \end{array} \]

Alternative 3: 98.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{x \cdot x} - -1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (if (<= x 2.9) (fma 0.225 (* x x) -0.5) (- (/ 3.0 (* x x)) -1.0)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 2.9) {
		tmp = fma(0.225, (x * x), -0.5);
	} else {
		tmp = (3.0 / (x * x)) - -1.0;
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 2.9)
		tmp = fma(0.225, Float64(x * x), -0.5);
	else
		tmp = Float64(Float64(3.0 / Float64(x * x)) - -1.0);
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 2.9], N[(0.225 * N[(x * x), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(3.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.89999999999999991

   1. Initial program 34.7%

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

      1. sub-neg 34.7%

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

      2. +-commutative 34.7%

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

      3. neg-sub0 34.7%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 34.7%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 34.7%

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

      6. neg-mul-1 34.7%

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

      7. sub-neg 34.7%

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

      8. +-commutative 34.7%

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

      9. neg-sub0 34.7%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 34.7%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 34.7%

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

      12. neg-mul-1 34.7%

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

      13. times-frac 34.7%

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

      14. metadata-eval 34.7%

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

      15. *-lft-identity 34.7%

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

   3. Simplified 34.7%

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

   4. Taylor expanded in x around 0 67.1%

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

      1. fma-neg 67.1%

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

      2. unpow2 67.1%

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

      3. metadata-eval 67.1%

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

   6. Simplified 67.1%

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

   if 2.89999999999999991 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 100.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

      1. div-sub 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{\frac{3}{{x}^{2}}} - \frac{x}{\tan x - x} \]
   7. Step-by-step derivation

      1. unpow2 99.2%

         \[\leadsto \frac{3}{\color{blue}{x \cdot x}} - \frac{x}{\tan x - x} \]

   8. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{3}{x \cdot x}} - \frac{x}{\tan x - x} \]

   9. Taylor expanded in x around inf 99.2%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.9:\\ \;\;\;\;\mathsf{fma}\left(0.225, x \cdot x, -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{x \cdot x} - -1\\ \end{array} \]

Alternative 4: 98.5% accurate, 22.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{x \cdot x} - -1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.6) -0.5 (- (/ 3.0 (* x x)) -1.0)))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = -0.5;
	} else {
		tmp = (3.0 / (x * x)) - -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 <= 1.6d0) then
        tmp = -0.5d0
    else
        tmp = (3.0d0 / (x * x)) - (-1.0d0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 1.6) {
		tmp = -0.5;
	} else {
		tmp = (3.0 / (x * x)) - -1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = -0.5
	else:
		tmp = (3.0 / (x * x)) - -1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.6)
		tmp = -0.5;
	else
		tmp = Float64(Float64(3.0 / Float64(x * x)) - -1.0);
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 1.6)
		tmp = -0.5;
	else
		tmp = (3.0 / (x * x)) - -1.0;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := If[LessEqual[x, 1.6], -0.5, N[(N[(3.0 / N[(x * x), $MachinePrecision]), $MachinePrecision] - -1.0), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 34.7%

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

      1. sub-neg 34.7%

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

      2. +-commutative 34.7%

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

      3. neg-sub0 34.7%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 34.7%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 34.7%

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

      6. neg-mul-1 34.7%

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

      7. sub-neg 34.7%

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

      8. +-commutative 34.7%

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

      9. neg-sub0 34.7%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 34.7%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 34.7%

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

      12. neg-mul-1 34.7%

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

      13. times-frac 34.7%

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

      14. metadata-eval 34.7%

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

      15. *-lft-identity 34.7%

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

   3. Simplified 34.7%

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

   4. Taylor expanded in x around 0 66.0%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 100.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

      1. div-sub 99.9%

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

   5. Applied egg-rr 99.9%

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

   6. Taylor expanded in x around 0 99.2%

      \[\leadsto \color{blue}{\frac{3}{{x}^{2}}} - \frac{x}{\tan x - x} \]
   7. Step-by-step derivation

      1. unpow2 99.2%

         \[\leadsto \frac{3}{\color{blue}{x \cdot x}} - \frac{x}{\tan x - x} \]

   8. Simplified 99.2%

      \[\leadsto \color{blue}{\frac{3}{x \cdot x}} - \frac{x}{\tan x - x} \]

   9. Taylor expanded in x around inf 99.2%

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

4. Final simplification 73.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.6:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{3}{x \cdot x} - -1\\ \end{array} \]

Alternative 5: 98.5% accurate, 67.6× speedup

Math

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

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.6) -0.5 1.0))

C

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

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.6:
		tmp = -0.5
	else:
		tmp = 1.0
	return tmp

Julia

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

Derivation

1. Split input into 2 regimes

2. if x < 1.6000000000000001

   1. Initial program 34.7%

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

      1. sub-neg 34.7%

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

      2. +-commutative 34.7%

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

      3. neg-sub0 34.7%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 34.7%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 34.7%

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

      6. neg-mul-1 34.7%

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

      7. sub-neg 34.7%

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

      8. +-commutative 34.7%

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

      9. neg-sub0 34.7%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 34.7%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 34.7%

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

      12. neg-mul-1 34.7%

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

      13. times-frac 34.7%

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

      14. metadata-eval 34.7%

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

      15. *-lft-identity 34.7%

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

   3. Simplified 34.7%

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

   4. Taylor expanded in x around 0 66.0%

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

   if 1.6000000000000001 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

      4. associate-+l- 100.0%

         \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

      5. sub0-neg 100.0%

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

      6. neg-mul-1 100.0%

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

      7. sub-neg 100.0%

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

      8. +-commutative 100.0%

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

      9. neg-sub0 100.0%

         \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

      10. associate-+l- 100.0%

          \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

      11. sub0-neg 100.0%

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

      12. neg-mul-1 100.0%

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

      13. times-frac 100.0%

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

      14. metadata-eval 100.0%

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

      15. *-lft-identity 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 99.2%

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

4. Final simplification 73.0%

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

Alternative 6: 51.0% 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 48.5%

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

   1. sub-neg 48.5%

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

   2. +-commutative 48.5%

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

   3. neg-sub0 48.5%

      \[\leadsto \frac{\color{blue}{\left(0 - \sin x\right)} + x}{x - \tan x} \]

   4. associate-+l- 48.5%

      \[\leadsto \frac{\color{blue}{0 - \left(\sin x - x\right)}}{x - \tan x} \]

   5. sub0-neg 48.5%

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

   6. neg-mul-1 48.5%

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

   7. sub-neg 48.5%

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

   8. +-commutative 48.5%

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

   9. neg-sub0 48.5%

      \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{\left(0 - \tan x\right)} + x} \]

   10. associate-+l- 48.5%

       \[\leadsto \frac{-1 \cdot \left(\sin x - x\right)}{\color{blue}{0 - \left(\tan x - x\right)}} \]

   11. sub0-neg 48.5%

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

   12. neg-mul-1 48.5%

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

   13. times-frac 48.5%

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

   14. metadata-eval 48.5%

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

   15. *-lft-identity 48.5%

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

3. Simplified 48.5%

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

4. Taylor expanded in x around 0 52.4%

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

5. Final simplification 52.4%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023230 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))