bug500 (missed optimization)

Percentage Accurate: 69.5% → 98.1%

Time: 7.7s

Alternatives: 4

Speedup: 1.0×

Specification

\[-1000 < x \land x < 1000\]

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 69.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{{x}^{3}}{-6} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (/ (pow x 3.0) -6.0))

C

double code(double x) {
	return pow(x, 3.0) / -6.0;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 3.0d0) / (-6.0d0)
end function

Java

public static double code(double x) {
	return Math.pow(x, 3.0) / -6.0;
}

Python

def code(x):
	return math.pow(x, 3.0) / -6.0

Julia

function code(x)
	return Float64((x ^ 3.0) / -6.0)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 3.0) / -6.0;
end

Wolfram

code[x_] := N[(N[Power[x, 3.0], $MachinePrecision] / -6.0), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.0%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3}} \]
4. Step-by-step derivation

   1. add-cube-cbrt 98.5%

      \[\leadsto \color{blue}{\left(\sqrt[3]{-0.16666666666666666 \cdot {x}^{3}} \cdot \sqrt[3]{-0.16666666666666666 \cdot {x}^{3}}\right) \cdot \sqrt[3]{-0.16666666666666666 \cdot {x}^{3}}} \]

   2. pow3 98.5%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{-0.16666666666666666 \cdot {x}^{3}}\right)}^{3}} \]

   3. *-commutative 98.5%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{{x}^{3} \cdot -0.16666666666666666}}\right)}^{3} \]

   4. cbrt-prod 98.0%

      \[\leadsto {\color{blue}{\left(\sqrt[3]{{x}^{3}} \cdot \sqrt[3]{-0.16666666666666666}\right)}}^{3} \]

   5. rem-cbrt-cube 98.1%

      \[\leadsto {\left(\color{blue}{x} \cdot \sqrt[3]{-0.16666666666666666}\right)}^{3} \]

5. Applied egg-rr 98.1%

   \[\leadsto \color{blue}{{\left(x \cdot \sqrt[3]{-0.16666666666666666}\right)}^{3}} \]
6. Step-by-step derivation

   1. unpow-prod-down 98.1%

      \[\leadsto \color{blue}{{x}^{3} \cdot {\left(\sqrt[3]{-0.16666666666666666}\right)}^{3}} \]

   2. rem-cube-cbrt 99.0%

      \[\leadsto {x}^{3} \cdot \color{blue}{-0.16666666666666666} \]

   3. metadata-eval 99.0%

      \[\leadsto {x}^{3} \cdot \color{blue}{\frac{1}{-6}} \]

   4. div-inv 99.1%

      \[\leadsto \color{blue}{\frac{{x}^{3}}{-6}} \]

7. Applied egg-rr 99.1%

   \[\leadsto \color{blue}{\frac{{x}^{3}}{-6}} \]

8. Final simplification 99.1%

   \[\leadsto \frac{{x}^{3}}{-6} \]
9. Add Preprocessing

Alternative 2: 98.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ {x}^{3} \cdot -0.16666666666666666 \end{array} \]

FPCore

(FPCore (x) :precision binary64 (* (pow x 3.0) -0.16666666666666666))

C

double code(double x) {
	return pow(x, 3.0) * -0.16666666666666666;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = (x ** 3.0d0) * (-0.16666666666666666d0)
end function

Java

public static double code(double x) {
	return Math.pow(x, 3.0) * -0.16666666666666666;
}

Python

def code(x):
	return math.pow(x, 3.0) * -0.16666666666666666

Julia

function code(x)
	return Float64((x ^ 3.0) * -0.16666666666666666)
end

MATLAB

function tmp = code(x)
	tmp = (x ^ 3.0) * -0.16666666666666666;
end

Wolfram

code[x_] := N[(N[Power[x, 3.0], $MachinePrecision] * -0.16666666666666666), $MachinePrecision]

Derivation

1. Initial program 68.5%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around 0 99.0%

   \[\leadsto \color{blue}{-0.16666666666666666 \cdot {x}^{3}} \]

4. Final simplification 99.0%

   \[\leadsto {x}^{3} \cdot -0.16666666666666666 \]
5. Add Preprocessing

Alternative 3: 69.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 68.5%

   \[\sin x - x \]
2. Add Preprocessing

3. Final simplification 68.5%

   \[\leadsto \sin x - x \]
4. Add Preprocessing

Alternative 4: 6.5% accurate, 51.5× speedup

Math

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

FPCore

(FPCore (x) :precision binary64 (- x))

C

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

Fortran

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

Java

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

Python

def code(x):
	return -x

Julia

function code(x)
	return Float64(-x)
end

MATLAB

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

Wolfram

code[x_] := (-x)

Derivation

1. Initial program 68.5%

   \[\sin x - x \]
2. Add Preprocessing

3. Taylor expanded in x around inf 6.3%

   \[\leadsto \color{blue}{-1 \cdot x} \]
4. Step-by-step derivation

   1. neg-mul-1 6.3%

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

5. Simplified 6.3%

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

6. Final simplification 6.3%

   \[\leadsto -x \]
7. Add Preprocessing

Developer target: 99.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| < 0.07:\\ \;\;\;\;-\left(\left(\frac{{x}^{3}}{6} - \frac{{x}^{5}}{120}\right) + \frac{{x}^{7}}{5040}\right)\\ \mathbf{else}:\\ \;\;\;\;\sin x - x\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (< (fabs x) 0.07)
   (- (+ (- (/ (pow x 3.0) 6.0) (/ (pow x 5.0) 120.0)) (/ (pow x 7.0) 5040.0)))
   (- (sin x) x)))

C

double code(double x) {
	double tmp;
	if (fabs(x) < 0.07) {
		tmp = -(((pow(x, 3.0) / 6.0) - (pow(x, 5.0) / 120.0)) + (pow(x, 7.0) / 5040.0));
	} else {
		tmp = sin(x) - x;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) < 0.07d0) then
        tmp = -((((x ** 3.0d0) / 6.0d0) - ((x ** 5.0d0) / 120.0d0)) + ((x ** 7.0d0) / 5040.0d0))
    else
        tmp = sin(x) - x
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) < 0.07) {
		tmp = -(((Math.pow(x, 3.0) / 6.0) - (Math.pow(x, 5.0) / 120.0)) + (Math.pow(x, 7.0) / 5040.0));
	} else {
		tmp = Math.sin(x) - x;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) < 0.07:
		tmp = -(((math.pow(x, 3.0) / 6.0) - (math.pow(x, 5.0) / 120.0)) + (math.pow(x, 7.0) / 5040.0))
	else:
		tmp = math.sin(x) - x
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) < 0.07)
		tmp = Float64(-Float64(Float64(Float64((x ^ 3.0) / 6.0) - Float64((x ^ 5.0) / 120.0)) + Float64((x ^ 7.0) / 5040.0)));
	else
		tmp = Float64(sin(x) - x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) < 0.07)
		tmp = -((((x ^ 3.0) / 6.0) - ((x ^ 5.0) / 120.0)) + ((x ^ 7.0) / 5040.0));
	else
		tmp = sin(x) - x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[Less[N[Abs[x], $MachinePrecision], 0.07], (-N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] / 6.0), $MachinePrecision] - N[(N[Power[x, 5.0], $MachinePrecision] / 120.0), $MachinePrecision]), $MachinePrecision] + N[(N[Power[x, 7.0], $MachinePrecision] / 5040.0), $MachinePrecision]), $MachinePrecision]), N[(N[Sin[x], $MachinePrecision] - x), $MachinePrecision]]

Reproduce

herbie shell --seed 2024026 
(FPCore (x)
  :name "bug500 (missed optimization)"
  :precision binary64
  :pre (and (< -1000.0 x) (< x 1000.0))

  :herbie-target
  (if (< (fabs x) 0.07) (- (+ (- (/ (pow x 3.0) 6.0) (/ (pow x 5.0) 120.0)) (/ (pow x 7.0) 5040.0))) (- (sin x) x))

  (- (sin x) x))