sintan (problem 3.4.5)

Percentage Accurate: 50.4% → 99.7%

Time: 13.2s

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: 99.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := x - \sin x\\ t_1 := x - \tan x\\ \mathbf{if}\;\frac{t_0}{t_1} \leq 2:\\ \;\;\;\;{\left(\frac{t_1}{t_0}\right)}^{-1}\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (- x (sin x))) (t_1 (- x (tan x))))
   (if (<= (/ t_0 t_1) 2.0) (pow (/ t_1 t_0) -1.0) -0.5)))

C

double code(double x) {
	double t_0 = x - sin(x);
	double t_1 = x - tan(x);
	double tmp;
	if ((t_0 / t_1) <= 2.0) {
		tmp = pow((t_1 / t_0), -1.0);
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: t_1
    real(8) :: tmp
    t_0 = x - sin(x)
    t_1 = x - tan(x)
    if ((t_0 / t_1) <= 2.0d0) then
        tmp = (t_1 / t_0) ** (-1.0d0)
    else
        tmp = -0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = x - Math.sin(x);
	double t_1 = x - Math.tan(x);
	double tmp;
	if ((t_0 / t_1) <= 2.0) {
		tmp = Math.pow((t_1 / t_0), -1.0);
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Python

def code(x):
	t_0 = x - math.sin(x)
	t_1 = x - math.tan(x)
	tmp = 0
	if (t_0 / t_1) <= 2.0:
		tmp = math.pow((t_1 / t_0), -1.0)
	else:
		tmp = -0.5
	return tmp

Julia

function code(x)
	t_0 = Float64(x - sin(x))
	t_1 = Float64(x - tan(x))
	tmp = 0.0
	if (Float64(t_0 / t_1) <= 2.0)
		tmp = Float64(t_1 / t_0) ^ -1.0;
	else
		tmp = -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = x - sin(x);
	t_1 = x - tan(x);
	tmp = 0.0;
	if ((t_0 / t_1) <= 2.0)
		tmp = (t_1 / t_0) ^ -1.0;
	else
		tmp = -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$0 / t$95$1), $MachinePrecision], 2.0], N[Power[N[(t$95$1 / t$95$0), $MachinePrecision], -1.0], $MachinePrecision], -0.5]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x))) < 2

   1. Initial program 100.0%

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

      1. clear-num 100.0%

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

      2. inv-pow 100.0%

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

   3. Applied egg-rr 100.0%

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

   if 2 < (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x)))

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - \sin x}{x - \tan x}\\ \mathbf{if}\;t_0 \leq 2:\\ \;\;\;\;t_0\\ \mathbf{else}:\\ \;\;\;\;-0.5\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ (- x (sin x)) (- x (tan x))))) (if (<= t_0 2.0) t_0 -0.5)))

C

double code(double x) {
	double t_0 = (x - sin(x)) / (x - tan(x));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - sin(x)) / (x - tan(x))
    if (t_0 <= 2.0d0) then
        tmp = t_0
    else
        tmp = -0.5d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double t_0 = (x - Math.sin(x)) / (x - Math.tan(x));
	double tmp;
	if (t_0 <= 2.0) {
		tmp = t_0;
	} else {
		tmp = -0.5;
	}
	return tmp;
}

Python

def code(x):
	t_0 = (x - math.sin(x)) / (x - math.tan(x))
	tmp = 0
	if t_0 <= 2.0:
		tmp = t_0
	else:
		tmp = -0.5
	return tmp

Julia

function code(x)
	t_0 = Float64(Float64(x - sin(x)) / Float64(x - tan(x)))
	tmp = 0.0
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = -0.5;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	t_0 = (x - sin(x)) / (x - tan(x));
	tmp = 0.0;
	if (t_0 <= 2.0)
		tmp = t_0;
	else
		tmp = -0.5;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := Block[{t$95$0 = N[(N[(x - N[Sin[x], $MachinePrecision]), $MachinePrecision] / N[(x - N[Tan[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$0, 2.0], t$95$0, -0.5]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x))) < 2

   1. Initial program 100.0%

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

   if 2 < (/.f64 (-.f64 x (sin.f64 x)) (-.f64 x (tan.f64 x)))

   1. Initial program 0.0%

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

   2. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{-0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 100.0%

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

Alternative 3: 75.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;{\left(x \cdot \sqrt{0.225}\right)}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.4)
   (- (pow (* x (sqrt 0.225)) 2.0) 0.5)
   (+ 1.0 (/ (- (tan x) (sin x)) x))))

C

double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = pow((x * sqrt(0.225)), 2.0) - 0.5;
	} else {
		tmp = 1.0 + ((tan(x) - sin(x)) / x);
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.4d0) then
        tmp = ((x * sqrt(0.225d0)) ** 2.0d0) - 0.5d0
    else
        tmp = 1.0d0 + ((tan(x) - sin(x)) / x)
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.4) {
		tmp = Math.pow((x * Math.sqrt(0.225)), 2.0) - 0.5;
	} else {
		tmp = 1.0 + ((Math.tan(x) - Math.sin(x)) / x);
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.4:
		tmp = math.pow((x * math.sqrt(0.225)), 2.0) - 0.5
	else:
		tmp = 1.0 + ((math.tan(x) - math.sin(x)) / x)
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.4)
		tmp = Float64((Float64(x * sqrt(0.225)) ^ 2.0) - 0.5);
	else
		tmp = Float64(1.0 + Float64(Float64(tan(x) - sin(x)) / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.4)
		tmp = ((x * sqrt(0.225)) ^ 2.0) - 0.5;
	else
		tmp = 1.0 + ((tan(x) - sin(x)) / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.4], N[(N[Power[N[(x * N[Sqrt[0.225], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - 0.5), $MachinePrecision], N[(1.0 + N[(N[(N[Tan[x], $MachinePrecision] - N[Sin[x], $MachinePrecision]), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 2.39999999999999991

   1. Initial program 31.7%

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

   2. Taylor expanded in x around 0 69.6%

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

      1. add-sqr-sqrt 69.6%

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

      2. pow2 69.6%

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

      3. *-commutative 69.6%

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

      4. sqrt-prod 69.6%

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

      5. unpow2 69.6%

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

      6. sqrt-prod 29.1%

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

      7. add-sqr-sqrt 69.6%

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

   4. Applied egg-rr 69.6%

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

   if 2.39999999999999991 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 97.5%

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

      1. associate--l+ 97.5%

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

      2. sub-neg 97.5%

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

      3. *-lft-identity 97.5%

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

      4. metadata-eval 97.5%

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

      5. cancel-sign-sub-inv 97.5%

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

      6. distribute-lft-out-- 97.5%

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

      7. mul-1-neg 97.5%

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

      8. remove-double-neg 97.5%

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

      9. *-commutative 97.5%

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

      10. associate-/r* 97.5%

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

      11. div-sub 97.5%

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

      12. associate-*r/ 97.5%

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

   4. Simplified 97.5%

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

      1. expm1-log1p-u 97.5%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x \cdot \left(-1 - \frac{-1}{\cos x}\right)}{x}\right)\right)} \]

      2. expm1-udef 97.5%

         \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin x \cdot \left(-1 - \frac{-1}{\cos x}\right)}{x}\right)} - 1\right)} \]

      3. sub-neg 97.5%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{\sin x \cdot \color{blue}{\left(-1 + \left(-\frac{-1}{\cos x}\right)\right)}}{x}\right)} - 1\right) \]

      4. neg-mul-1 97.5%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{\sin x \cdot \left(-1 + \color{blue}{-1 \cdot \frac{-1}{\cos x}}\right)}{x}\right)} - 1\right) \]

      5. distribute-lft-in 97.5%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{\color{blue}{\sin x \cdot -1 + \sin x \cdot \left(-1 \cdot \frac{-1}{\cos x}\right)}}{x}\right)} - 1\right) \]

      6. associate-*r/ 97.5%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{\sin x \cdot -1 + \sin x \cdot \color{blue}{\frac{-1 \cdot -1}{\cos x}}}{x}\right)} - 1\right) \]

      7. metadata-eval 97.5%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{\sin x \cdot -1 + \sin x \cdot \frac{\color{blue}{1}}{\cos x}}{x}\right)} - 1\right) \]

      8. div-inv 97.5%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{\sin x \cdot -1 + \color{blue}{\frac{\sin x}{\cos x}}}{x}\right)} - 1\right) \]

      9. tan-quot 97.5%

         \[\leadsto 1 + \left(e^{\mathsf{log1p}\left(\frac{\sin x \cdot -1 + \color{blue}{\tan x}}{x}\right)} - 1\right) \]

   6. Applied egg-rr 97.5%

      \[\leadsto 1 + \color{blue}{\left(e^{\mathsf{log1p}\left(\frac{\sin x \cdot -1 + \tan x}{x}\right)} - 1\right)} \]
   7. Step-by-step derivation

      1. expm1-def 97.5%

         \[\leadsto 1 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\sin x \cdot -1 + \tan x}{x}\right)\right)} \]

      2. expm1-log1p 97.5%

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

      3. +-commutative 97.5%

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

      4. *-commutative 97.5%

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

      5. neg-mul-1 97.5%

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

      6. unsub-neg 97.5%

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

   8. Simplified 97.5%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.4:\\ \;\;\;\;{\left(x \cdot \sqrt{0.225}\right)}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1 + \frac{\tan x - \sin x}{x}\\ \end{array} \]

Alternative 4: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55:\\ \;\;\;\;{\left(x \cdot \sqrt{0.225}\right)}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= x 2.55) (- (pow (* x (sqrt 0.225)) 2.0) 0.5) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 2.55) {
		tmp = pow((x * sqrt(0.225)), 2.0) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.55d0) then
        tmp = ((x * sqrt(0.225d0)) ** 2.0d0) - 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.55) {
		tmp = Math.pow((x * Math.sqrt(0.225)), 2.0) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.55:
		tmp = math.pow((x * math.sqrt(0.225)), 2.0) - 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.55)
		tmp = Float64((Float64(x * sqrt(0.225)) ^ 2.0) - 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.55)
		tmp = ((x * sqrt(0.225)) ^ 2.0) - 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.55], N[(N[Power[N[(x * N[Sqrt[0.225], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] - 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.5499999999999998

   1. Initial program 31.7%

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

   2. Taylor expanded in x around 0 69.6%

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

      1. add-sqr-sqrt 69.6%

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

      2. pow2 69.6%

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

      3. *-commutative 69.6%

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

      4. sqrt-prod 69.6%

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

      5. unpow2 69.6%

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

      6. sqrt-prod 29.1%

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

      7. add-sqr-sqrt 69.6%

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

   4. Applied egg-rr 69.6%

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

   if 2.5499999999999998 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 95.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.55:\\ \;\;\;\;{\left(x \cdot \sqrt{0.225}\right)}^{2} - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 5: 75.5% accurate, 22.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 2.55:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= x 2.55) (- (* x (* x 0.225)) 0.5) 1.0))

C

double code(double x) {
	double tmp;
	if (x <= 2.55) {
		tmp = (x * (x * 0.225)) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 2.55d0) then
        tmp = (x * (x * 0.225d0)) - 0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (x <= 2.55) {
		tmp = (x * (x * 0.225)) - 0.5;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if x <= 2.55:
		tmp = (x * (x * 0.225)) - 0.5
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (x <= 2.55)
		tmp = Float64(Float64(x * Float64(x * 0.225)) - 0.5);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (x <= 2.55)
		tmp = (x * (x * 0.225)) - 0.5;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[x, 2.55], N[(N[(x * N[(x * 0.225), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 2.5499999999999998

   1. Initial program 31.7%

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

   2. Taylor expanded in x around 0 69.6%

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

      1. add-sqr-sqrt 69.6%

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

      2. pow2 69.6%

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

      3. *-commutative 69.6%

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

      4. sqrt-prod 69.6%

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

      5. unpow2 69.6%

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

      6. sqrt-prod 29.1%

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

      7. add-sqr-sqrt 69.6%

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

   4. Applied egg-rr 69.6%

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

      1. unpow2 69.6%

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

      2. swap-sqr 69.6%

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

      3. rem-square-sqrt 69.6%

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

      4. associate-*l* 69.6%

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

   6. Applied egg-rr 69.6%

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

   if 2.5499999999999998 < x

   1. Initial program 100.0%

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

   2. Taylor expanded in x around inf 95.2%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2.55:\\ \;\;\;\;x \cdot \left(x \cdot 0.225\right) - 0.5\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 74.6% accurate, 67.6× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (if (<= x 1.55) -0.5 1.0))

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (x <= 1.55d0) then
        tmp = -0.5d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_] := If[LessEqual[x, 1.55], -0.5, 1.0]

Derivation

1. Split input into 2 regimes

2. if x < 1.55000000000000004

   1. Initial program 31.4%

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

   2. Taylor expanded in x around 0 69.1%

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

   if 1.55000000000000004 < x

   1. Initial program 99.9%

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

   2. Taylor expanded in x around inf 94.0%

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

4. Final simplification 76.0%

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

Alternative 7: 50.7% accurate, 207.0× speedup

Math

\[\begin{array}{l} \\ -0.5 \end{array} \]

FPCore

(FPCore (x) :precision binary64 -0.5)

C

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

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = -0.5d0
end function

Java

public static double code(double x) {
	return -0.5;
}

Python

def code(x):
	return -0.5

Julia

function code(x)
	return -0.5
end

MATLAB

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

Wolfram

code[x_] := -0.5

Derivation

1. Initial program 50.4%

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

2. Taylor expanded in x around 0 50.4%

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

3. Final simplification 50.4%

   \[\leadsto -0.5 \]

Reproduce

herbie shell --seed 2023333 
(FPCore (x)
  :name "sintan (problem 3.4.5)"
  :precision binary64
  (/ (- x (sin x)) (- x (tan x))))