sintan (problem 3.4.5)

Percentage Accurate: 51.3% → 99.9%

Time: 19.4s

Alternatives: 7

Speedup: 67.6×

Specification

Math

\[\begin{array}{l} \\ \frac{x - \sin x}{x - \tan x} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (- x (sin x)) (- x (tan x))))

C

double code(double x) {
	return (x - sin(x)) / (x - tan(x));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x - sin(x)) / (x - tan(x))
end function

Java

public static double code(double x) {
	return (x - Math.sin(x)) / (x - Math.tan(x));
}

Python

def code(x):
	return (x - math.sin(x)) / (x - math.tan(x))

Julia

function code(x)
	return Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
end

MATLAB

function tmp = code(x)
	tmp = (x - sin(x)) / (x - tan(x));
end

Wolfram

code[x_] := N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 51.3% 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, 0.3× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \sqrt[3]{\sin x}\\ \mathbf{if}\;x \leq 0.0066:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({t_0}^{2}, t_0, -x\right)}{\tan x - x}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (cbrt (sin x))))
   (if (<= x 0.0066)
     (+ (* 0.225 (* x x)) -0.5)
     (/ (fma (pow t_0 2.0) t_0 (- x)) (- (tan x) x)))))

C

x = abs(x);
double code(double x) {
	double t_0 = cbrt(sin(x));
	double tmp;
	if (x <= 0.0066) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = fma(pow(t_0, 2.0), t_0, -x) / (tan(x) - x);
	}
	return tmp;
}

Julia

x = abs(x)
function code(x)
	t_0 = cbrt(sin(x))
	tmp = 0.0
	if (x <= 0.0066)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(fma((t_0 ^ 2.0), t_0, Float64(-x)) / Float64(tan(x) - x));
	end
	return tmp
end

Wolfram

NOTE: x should be positive before calling this function
code[x_] := Block[{t$95$0 = N[Power[N[Sin[x], $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[x, 0.0066], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(N[Power[t$95$0, 2.0], $MachinePrecision] * t$95$0 + (-x)), $MachinePrecision] / N[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.0066

   1. Initial program 36.5%

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

      1. sub-neg 36.5%

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

      2. +-commutative 36.5%

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

      3. neg-sub0 36.5%

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

      4. associate-+l- 36.5%

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

      5. sub0-neg 36.5%

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

      6. neg-mul-1 36.5%

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

      7. sub-neg 36.5%

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

      8. +-commutative 36.5%

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

      9. neg-sub0 36.5%

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

      10. associate-+l- 36.5%

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

      11. sub0-neg 36.5%

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

      12. neg-mul-1 36.5%

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

      13. times-frac 36.5%

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

      14. metadata-eval 36.5%

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

      15. *-lft-identity 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in x around 0 65.6%

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

      1. fma-neg 65.6%

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

      2. unpow2 65.6%

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

      3. metadata-eval 65.6%

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

   6. Simplified 65.6%

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

      1. fma-udef 65.6%

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

   8. Applied egg-rr 65.6%

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

   if 0.0066 < 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. add-cube-cbrt 100.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}} - x}{\tan x - x} \]

      2. fma-neg 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x}, -x\right)}}{\tan x - x} \]

      3. pow2 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sin x}\right)}^{2}}, \sqrt[3]{\sin x}, -x\right)}{\tan x - x} \]

   5. Applied egg-rr 100.0%

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

4. Final simplification 74.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 0.0066:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left({\left(\sqrt[3]{\sin x}\right)}^{2}, \sqrt[3]{\sin x}, -x\right)}{\tan x - x}\\ \end{array} \]

Alternative 2: 99.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} t_0 := \tan x - x\\ \mathbf{if}\;x \leq 0.0048:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;\frac{\sin x}{t_0} - \frac{x}{t_0}\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x)
 :precision binary64
 (let* ((t_0 (- (tan x) x)))
   (if (<= x 0.0048)
     (+ (* 0.225 (* x x)) -0.5)
     (- (/ (sin x) t_0) (/ x t_0)))))

C

x = abs(x);
double code(double x) {
	double t_0 = tan(x) - x;
	double tmp;
	if (x <= 0.0048) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (sin(x) / t_0) - (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 = tan(x) - x
    if (x <= 0.0048d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = (sin(x) / t_0) - (x / t_0)
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double t_0 = Math.tan(x) - x;
	double tmp;
	if (x <= 0.0048) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = (Math.sin(x) / t_0) - (x / t_0);
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	t_0 = math.tan(x) - x
	tmp = 0
	if x <= 0.0048:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = (math.sin(x) / t_0) - (x / t_0)
	return tmp

Julia

x = abs(x)
function code(x)
	t_0 = Float64(tan(x) - x)
	tmp = 0.0
	if (x <= 0.0048)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = Float64(Float64(sin(x) / t_0) - Float64(x / t_0));
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	t_0 = tan(x) - x;
	tmp = 0.0;
	if (x <= 0.0048)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = (sin(x) / t_0) - (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[(N[Tan[x], $MachinePrecision] - x), $MachinePrecision]}, If[LessEqual[x, 0.0048], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], N[(N[(N[Sin[x], $MachinePrecision] / t$95$0), $MachinePrecision] - N[(x / t$95$0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < 0.00479999999999999958

   1. Initial program 36.5%

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

      1. sub-neg 36.5%

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

      2. +-commutative 36.5%

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

      3. neg-sub0 36.5%

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

      4. associate-+l- 36.5%

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

      5. sub0-neg 36.5%

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

      6. neg-mul-1 36.5%

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

      7. sub-neg 36.5%

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

      8. +-commutative 36.5%

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

      9. neg-sub0 36.5%

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

      10. associate-+l- 36.5%

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

      11. sub0-neg 36.5%

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

      12. neg-mul-1 36.5%

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

      13. times-frac 36.5%

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

      14. metadata-eval 36.5%

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

      15. *-lft-identity 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in x around 0 65.6%

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

      1. fma-neg 65.6%

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

      2. unpow2 65.6%

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

      3. metadata-eval 65.6%

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

   6. Simplified 65.6%

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

      1. fma-udef 65.6%

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

   8. Applied egg-rr 65.6%

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

   if 0.00479999999999999958 < 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 100.0%

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

   5. Applied egg-rr 100.0%

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

4. Final simplification 74.8%

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

Alternative 3: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = 1.0 / ((tan(x) - x) / (sin(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 <= 0.0044d0) then
        tmp = (0.225d0 * (x * x)) + (-0.5d0)
    else
        tmp = 1.0d0 / ((tan(x) - x) / (sin(x) - x))
    end if
    code = tmp
end function

Java

x = Math.abs(x);
public static double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = 1.0 / ((Math.tan(x) - x) / (Math.sin(x) - x));
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 0.0044:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = 1.0 / ((math.tan(x) - x) / (math.sin(x) - x))
	return tmp

Julia

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

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 0.0044)
		tmp = (0.225 * (x * x)) + -0.5;
	else
		tmp = 1.0 / ((tan(x) - x) / (sin(x) - x));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if x < 0.00440000000000000027

   1. Initial program 36.5%

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

      1. sub-neg 36.5%

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

      2. +-commutative 36.5%

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

      3. neg-sub0 36.5%

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

      4. associate-+l- 36.5%

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

      5. sub0-neg 36.5%

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

      6. neg-mul-1 36.5%

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

      7. sub-neg 36.5%

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

      8. +-commutative 36.5%

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

      9. neg-sub0 36.5%

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

      10. associate-+l- 36.5%

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

      11. sub0-neg 36.5%

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

      12. neg-mul-1 36.5%

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

      13. times-frac 36.5%

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

      14. metadata-eval 36.5%

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

      15. *-lft-identity 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in x around 0 65.6%

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

      1. fma-neg 65.6%

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

      2. unpow2 65.6%

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

      3. metadata-eval 65.6%

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

   6. Simplified 65.6%

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

      1. fma-udef 65.6%

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

   8. Applied egg-rr 65.6%

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

   if 0.00440000000000000027 < 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. clear-num 100.0%

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

      2. associate-/r/ 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. add-cube-cbrt 100.0%

         \[\leadsto \frac{\color{blue}{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right) \cdot \sqrt[3]{\sin x}} - x}{\tan x - x} \]

      2. fma-neg 100.0%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}, \sqrt[3]{\sin x}, -x\right)}}{\tan x - x} \]

      3. pow2 100.0%

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{{\left(\sqrt[3]{\sin x}\right)}^{2}}, \sqrt[3]{\sin x}, -x\right)}{\tan x - x} \]

   7. Applied egg-rr 99.7%

      \[\leadsto \frac{1}{\tan x - x} \cdot \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\sin x}\right)}^{2}, \sqrt[3]{\sin x}, -x\right)} \]
   8. Step-by-step derivation

      1. *-commutative 99.7%

         \[\leadsto \color{blue}{\mathsf{fma}\left({\left(\sqrt[3]{\sin x}\right)}^{2}, \sqrt[3]{\sin x}, -x\right) \cdot \frac{1}{\tan x - x}} \]

      2. fma-udef 99.7%

         \[\leadsto \color{blue}{\left({\left(\sqrt[3]{\sin x}\right)}^{2} \cdot \sqrt[3]{\sin x} + \left(-x\right)\right)} \cdot \frac{1}{\tan x - x} \]

      3. unpow2 99.7%

         \[\leadsto \left(\color{blue}{\left(\sqrt[3]{\sin x} \cdot \sqrt[3]{\sin x}\right)} \cdot \sqrt[3]{\sin x} + \left(-x\right)\right) \cdot \frac{1}{\tan x - x} \]

      4. add-cube-cbrt 99.7%

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

      5. sub-neg 99.7%

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

      6. pow1 99.7%

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

      7. metadata-eval 99.7%

         \[\leadsto \left(\sin x - x\right) \cdot \frac{1}{{\tan x}^{\color{blue}{\left(3 \cdot 0.3333333333333333\right)}} - x} \]

      8. pow-pow 49.9%

         \[\leadsto \left(\sin x - x\right) \cdot \frac{1}{\color{blue}{{\left({\tan x}^{3}\right)}^{0.3333333333333333}} - x} \]

      9. div-inv 50.0%

         \[\leadsto \color{blue}{\frac{\sin x - x}{{\left({\tan x}^{3}\right)}^{0.3333333333333333} - x}} \]

      10. clear-num 50.0%

          \[\leadsto \color{blue}{\frac{1}{\frac{{\left({\tan x}^{3}\right)}^{0.3333333333333333} - x}{\sin x - x}}} \]

      11. pow-pow 100.0%

          \[\leadsto \frac{1}{\frac{\color{blue}{{\tan x}^{\left(3 \cdot 0.3333333333333333\right)}} - x}{\sin x - x}} \]

      12. metadata-eval 100.0%

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

      13. pow1 100.0%

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

   9. Applied egg-rr 100.0%

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

4. Final simplification 74.8%

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

Alternative 4: 99.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 0.0044) {
		tmp = (0.225 * (x * x)) + -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 <= 0.0044d0) then
        tmp = (0.225d0 * (x * x)) + (-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 <= 0.0044) {
		tmp = (0.225 * (x * x)) + -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 <= 0.0044:
		tmp = (0.225 * (x * x)) + -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 <= 0.0044)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -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 <= 0.0044)
		tmp = (0.225 * (x * x)) + -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, 0.0044], N[(N[(0.225 * N[(x * x), $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 < 0.00440000000000000027

   1. Initial program 36.5%

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

      1. sub-neg 36.5%

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

      2. +-commutative 36.5%

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

      3. neg-sub0 36.5%

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

      4. associate-+l- 36.5%

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

      5. sub0-neg 36.5%

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

      6. neg-mul-1 36.5%

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

      7. sub-neg 36.5%

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

      8. +-commutative 36.5%

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

      9. neg-sub0 36.5%

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

      10. associate-+l- 36.5%

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

      11. sub0-neg 36.5%

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

      12. neg-mul-1 36.5%

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

      13. times-frac 36.5%

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

      14. metadata-eval 36.5%

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

      15. *-lft-identity 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in x around 0 65.6%

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

      1. fma-neg 65.6%

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

      2. unpow2 65.6%

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

      3. metadata-eval 65.6%

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

   6. Simplified 65.6%

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

      1. fma-udef 65.6%

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

   8. Applied egg-rr 65.6%

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

   if 0.00440000000000000027 < x

   1. Initial program 100.0%

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

4. Final simplification 74.8%

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

Alternative 5: 99.0% accurate, 22.8× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 4.5) (+ (* 0.225 (* x x)) -0.5) 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 4.5) {
		tmp = (0.225 * (x * x)) + -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 <= 4.5d0) then
        tmp = (0.225d0 * (x * x)) + (-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 <= 4.5) {
		tmp = (0.225 * (x * x)) + -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 4.5:
		tmp = (0.225 * (x * x)) + -0.5
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 4.5)
		tmp = Float64(Float64(0.225 * Float64(x * x)) + -0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

x = abs(x)
function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 4.5)
		tmp = (0.225 * (x * x)) + -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, 4.5], N[(N[(0.225 * N[(x * x), $MachinePrecision]), $MachinePrecision] + -0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 4.5

   1. Initial program 36.5%

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

      1. sub-neg 36.5%

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

      2. +-commutative 36.5%

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

      3. neg-sub0 36.5%

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

      4. associate-+l- 36.5%

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

      5. sub0-neg 36.5%

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

      6. neg-mul-1 36.5%

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

      7. sub-neg 36.5%

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

      8. +-commutative 36.5%

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

      9. neg-sub0 36.5%

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

      10. associate-+l- 36.5%

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

      11. sub0-neg 36.5%

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

      12. neg-mul-1 36.5%

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

      13. times-frac 36.5%

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

      14. metadata-eval 36.5%

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

      15. *-lft-identity 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in x around 0 65.6%

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

      1. fma-neg 65.6%

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

      2. unpow2 65.6%

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

      3. metadata-eval 65.6%

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

   6. Simplified 65.6%

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

      1. fma-udef 65.6%

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

   8. Applied egg-rr 65.6%

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

   if 4.5 < 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 99.7%

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

4. Final simplification 74.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 4.5:\\ \;\;\;\;0.225 \cdot \left(x \cdot x\right) + -0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 98.6% accurate, 67.6× speedup

Math

\[\begin{array}{l} x = |x|\\ \\ \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

NOTE: x should be positive before calling this function
(FPCore (x) :precision binary64 (if (<= x 1.56) -0.5 1.0))

C

x = abs(x);
double code(double x) {
	double tmp;
	if (x <= 1.56) {
		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 <= 1.56d0) 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 <= 1.56) {
		tmp = -0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

x = abs(x)
def code(x):
	tmp = 0
	if x <= 1.56:
		tmp = -0.5
	else:
		tmp = 1.0
	return tmp

Julia

x = abs(x)
function code(x)
	tmp = 0.0
	if (x <= 1.56)
		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 <= 1.56)
		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, 1.56], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.5600000000000001

   1. Initial program 36.5%

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

      1. sub-neg 36.5%

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

      2. +-commutative 36.5%

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

      3. neg-sub0 36.5%

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

      4. associate-+l- 36.5%

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

      5. sub0-neg 36.5%

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

      6. neg-mul-1 36.5%

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

      7. sub-neg 36.5%

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

      8. +-commutative 36.5%

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

      9. neg-sub0 36.5%

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

      10. associate-+l- 36.5%

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

      11. sub0-neg 36.5%

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

      12. neg-mul-1 36.5%

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

      13. times-frac 36.5%

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

      14. metadata-eval 36.5%

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

      15. *-lft-identity 36.5%

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

   3. Simplified 36.5%

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

   4. Taylor expanded in x around 0 64.4%

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

   if 1.5600000000000001 < 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 99.7%

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

4. Final simplification 73.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.56:\\ \;\;\;\;-0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 7: 49.9% 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 53.3%

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

   1. sub-neg 53.3%

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

   2. +-commutative 53.3%

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

   3. neg-sub0 53.3%

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

   4. associate-+l- 53.3%

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

   5. sub0-neg 53.3%

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

   6. neg-mul-1 53.3%

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

   7. sub-neg 53.3%

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

   8. +-commutative 53.3%

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

   9. neg-sub0 53.3%

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

   10. associate-+l- 53.3%

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

   11. sub0-neg 53.3%

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

   12. neg-mul-1 53.3%

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

   13. times-frac 53.3%

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

   14. metadata-eval 53.3%

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

   15. *-lft-identity 53.3%

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

3. Simplified 53.3%

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

4. Taylor expanded in x around 0 47.7%

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

5. Final simplification 47.7%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023224 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))