Linear.Projection:inversePerspective from linear-1.19.1.3, C

Percentage Accurate: 76.8% → 100.0%

Time: 3.1s

Alternatives: 4

Speedup: 1.3×

Specification

Math

\[\begin{array}{l} \\ \frac{x + y}{\left(x \cdot 2\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (* (* x 2.0) y)))

C

double code(double x, double y) {
	return (x + y) / ((x * 2.0) * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / ((x * 2.0d0) * y)
end function

Java

public static double code(double x, double y) {
	return (x + y) / ((x * 2.0) * y);
}

Python

def code(x, y):
	return (x + y) / ((x * 2.0) * y)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(Float64(x * 2.0) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / ((x * 2.0) * y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Initial Program: 76.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x + y}{\left(x \cdot 2\right) \cdot y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ (+ x y) (* (* x 2.0) y)))

C

double code(double x, double y) {
	return (x + y) / ((x * 2.0) * y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / ((x * 2.0d0) * y)
end function

Java

public static double code(double x, double y) {
	return (x + y) / ((x * 2.0) * y);
}

Python

def code(x, y):
	return (x + y) / ((x * 2.0) * y)

Julia

function code(x, y)
	return Float64(Float64(x + y) / Float64(Float64(x * 2.0) * y))
end

MATLAB

function tmp = code(x, y)
	tmp = (x + y) / ((x * 2.0) * y);
end

Wolfram

code[x_, y_] := N[(N[(x + y), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{y} + \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	return (0.5 / y) + (0.5 / x);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.5d0 / y) + (0.5d0 / x)
end function

Java

public static double code(double x, double y) {
	return (0.5 / y) + (0.5 / x);
}

Python

def code(x, y):
	return (0.5 / y) + (0.5 / x)

Julia

function code(x, y)
	return Float64(Float64(0.5 / y) + Float64(0.5 / x))
end

MATLAB

function tmp = code(x, y)
	tmp = (0.5 / y) + (0.5 / x);
end

Wolfram

code[x_, y_] := N[(N[(0.5 / y), $MachinePrecision] + N[(0.5 / x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 78.8%

   \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]

2. Taylor expanded in x around 0 99.9%

   \[\leadsto \color{blue}{0.5 \cdot \frac{1}{y} + 0.5 \cdot \frac{1}{x}} \]
3. Step-by-step derivation

   1. associate-*r/ 99.9%

      \[\leadsto \color{blue}{\frac{0.5 \cdot 1}{y}} + 0.5 \cdot \frac{1}{x} \]

   2. metadata-eval 99.9%

      \[\leadsto \frac{\color{blue}{0.5}}{y} + 0.5 \cdot \frac{1}{x} \]

   3. associate-*r/ 99.9%

      \[\leadsto \frac{0.5}{y} + \color{blue}{\frac{0.5 \cdot 1}{x}} \]

   4. metadata-eval 99.9%

      \[\leadsto \frac{0.5}{y} + \frac{\color{blue}{0.5}}{x} \]

4. Simplified 99.9%

   \[\leadsto \color{blue}{\frac{0.5}{y} + \frac{0.5}{x}} \]

5. Final simplification 99.9%

   \[\leadsto \frac{0.5}{y} + \frac{0.5}{x} \]

Alternative 2: 61.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-56}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (if (<= y 1.65e-56) (/ 0.5 y) (/ 0.5 x)))

C

double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-56) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (y <= 1.65d-56) then
        tmp = 0.5d0 / y
    else
        tmp = 0.5d0 / x
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (y <= 1.65e-56) {
		tmp = 0.5 / y;
	} else {
		tmp = 0.5 / x;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if y <= 1.65e-56:
		tmp = 0.5 / y
	else:
		tmp = 0.5 / x
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (y <= 1.65e-56)
		tmp = Float64(0.5 / y);
	else
		tmp = Float64(0.5 / x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (y <= 1.65e-56)
		tmp = 0.5 / y;
	else
		tmp = 0.5 / x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[y, 1.65e-56], N[(0.5 / y), $MachinePrecision], N[(0.5 / x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 1.64999999999999992e-56

   1. Initial program 76.3%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]

   2. Taylor expanded in x around inf 62.5%

      \[\leadsto \color{blue}{\frac{0.5}{y}} \]

   if 1.64999999999999992e-56 < y

   1. Initial program 84.5%

      \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]

   2. Taylor expanded in x around 0 63.0%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 1.65 \cdot 10^{-56}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{x}\\ \end{array} \]

Alternative 3: 50.7% accurate, 3.0× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (/ 0.5 x))

C

double code(double x, double y) {
	return 0.5 / x;
}

Fortran

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

Java

public static double code(double x, double y) {
	return 0.5 / x;
}

Python

def code(x, y):
	return 0.5 / x

Julia

function code(x, y)
	return Float64(0.5 / x)
end

MATLAB

function tmp = code(x, y)
	tmp = 0.5 / x;
end

Wolfram

code[x_, y_] := N[(0.5 / x), $MachinePrecision]

Derivation

1. Initial program 78.8%

   \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]

2. Taylor expanded in x around 0 45.9%

   \[\leadsto \color{blue}{\frac{0.5}{x}} \]

3. Final simplification 45.9%

   \[\leadsto \frac{0.5}{x} \]

Alternative 4: 2.7% accurate, 9.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 0.0)

C

double code(double x, double y) {
	return 0.0;
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 0.0d0
end function

Java

public static double code(double x, double y) {
	return 0.0;
}

Python

def code(x, y):
	return 0.0

Julia

function code(x, y)
	return 0.0
end

MATLAB

function tmp = code(x, y)
	tmp = 0.0;
end

Wolfram

code[x_, y_] := 0.0

Derivation

1. Initial program 78.8%

   \[\frac{x + y}{\left(x \cdot 2\right) \cdot y} \]
2. Step-by-step derivation

   1. add-log-exp 7.8%

      \[\leadsto \color{blue}{\log \left(e^{\frac{x + y}{\left(x \cdot 2\right) \cdot y}}\right)} \]

   2. *-un-lft-identity 7.8%

      \[\leadsto \log \color{blue}{\left(1 \cdot e^{\frac{x + y}{\left(x \cdot 2\right) \cdot y}}\right)} \]

   3. log-prod 7.8%

      \[\leadsto \color{blue}{\log 1 + \log \left(e^{\frac{x + y}{\left(x \cdot 2\right) \cdot y}}\right)} \]

   4. metadata-eval 7.8%

      \[\leadsto \color{blue}{0} + \log \left(e^{\frac{x + y}{\left(x \cdot 2\right) \cdot y}}\right) \]

   5. add-log-exp 78.8%

      \[\leadsto 0 + \color{blue}{\frac{x + y}{\left(x \cdot 2\right) \cdot y}} \]

   6. associate-*l* 78.8%

      \[\leadsto 0 + \frac{x + y}{\color{blue}{x \cdot \left(2 \cdot y\right)}} \]

   7. *-commutative 78.8%

      \[\leadsto 0 + \frac{x + y}{x \cdot \color{blue}{\left(y \cdot 2\right)}} \]

3. Applied egg-rr 78.8%

   \[\leadsto \color{blue}{0 + \frac{x + y}{x \cdot \left(y \cdot 2\right)}} \]
4. Step-by-step derivation

   1. +-lft-identity 78.8%

      \[\leadsto \color{blue}{\frac{x + y}{x \cdot \left(y \cdot 2\right)}} \]

   2. associate-/r* 88.5%

      \[\leadsto \color{blue}{\frac{\frac{x + y}{x}}{y \cdot 2}} \]

5. Simplified 88.5%

   \[\leadsto \color{blue}{\frac{\frac{x + y}{x}}{y \cdot 2}} \]
6. Step-by-step derivation

   1. associate-/l/ 78.8%

      \[\leadsto \color{blue}{\frac{x + y}{\left(y \cdot 2\right) \cdot x}} \]

   2. associate-*l* 78.8%

      \[\leadsto \frac{x + y}{\color{blue}{y \cdot \left(2 \cdot x\right)}} \]

   3. *-commutative 78.8%

      \[\leadsto \frac{x + y}{y \cdot \color{blue}{\left(x \cdot 2\right)}} \]

   4. metadata-eval 78.8%

      \[\leadsto \frac{x + y}{y \cdot \left(x \cdot \color{blue}{\frac{1}{0.5}}\right)} \]

   5. div-inv 78.8%

      \[\leadsto \frac{x + y}{y \cdot \color{blue}{\frac{x}{0.5}}} \]

   6. *-commutative 78.8%

      \[\leadsto \frac{x + y}{\color{blue}{\frac{x}{0.5} \cdot y}} \]

   7. associate-/l/ 86.7%

      \[\leadsto \color{blue}{\frac{\frac{x + y}{y}}{\frac{x}{0.5}}} \]

   8. div-inv 86.7%

      \[\leadsto \frac{\frac{x + y}{y}}{\color{blue}{x \cdot \frac{1}{0.5}}} \]

   9. metadata-eval 86.7%

      \[\leadsto \frac{\frac{x + y}{y}}{x \cdot \color{blue}{2}} \]

   10. *-commutative 86.7%

       \[\leadsto \frac{\frac{x + y}{y}}{\color{blue}{2 \cdot x}} \]

   11. count-2 86.7%

       \[\leadsto \frac{\frac{x + y}{y}}{\color{blue}{x + x}} \]

   12. flip-+ 0.0%

       \[\leadsto \frac{\frac{x + y}{y}}{\color{blue}{\frac{x \cdot x - x \cdot x}{x - x}}} \]

   13. +-inverses 0.0%

       \[\leadsto \frac{\frac{x + y}{y}}{\frac{\color{blue}{0}}{x - x}} \]

   14. metadata-eval 0.0%

       \[\leadsto \frac{\frac{x + y}{y}}{\frac{\color{blue}{\log 1}}{x - x}} \]

   15. +-inverses 0.0%

       \[\leadsto \frac{\frac{x + y}{y}}{\frac{\log 1}{\color{blue}{0}}} \]

   16. metadata-eval 0.0%

       \[\leadsto \frac{\frac{x + y}{y}}{\frac{\log 1}{\color{blue}{\log 1}}} \]

   17. associate-/r/ 0.0%

       \[\leadsto \color{blue}{\frac{\frac{x + y}{y}}{\log 1} \cdot \log 1} \]

   18. +-commutative 0.0%

       \[\leadsto \frac{\frac{\color{blue}{y + x}}{y}}{\log 1} \cdot \log 1 \]

   19. metadata-eval 0.0%

       \[\leadsto \frac{\frac{y + x}{y}}{\color{blue}{0}} \cdot \log 1 \]

   20. metadata-eval 0.0%

       \[\leadsto \frac{\frac{y + x}{y}}{0} \cdot \color{blue}{0} \]

7. Applied egg-rr 0.0%

   \[\leadsto \color{blue}{\frac{\frac{y + x}{y}}{0} \cdot 0} \]
8. Step-by-step derivation

   1. mul0-rgt 2.7%

      \[\leadsto \color{blue}{0} \]

9. Simplified 2.7%

   \[\leadsto \color{blue}{0} \]

10. Final simplification 2.7%

    \[\leadsto 0 \]

Developer target: 100.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \frac{0.5}{x} + \frac{0.5}{y} \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (+ (/ 0.5 x) (/ 0.5 y)))

C

double code(double x, double y) {
	return (0.5 / x) + (0.5 / y);
}

Fortran

real(8) function code(x, y)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (0.5d0 / x) + (0.5d0 / y)
end function

Java

public static double code(double x, double y) {
	return (0.5 / x) + (0.5 / y);
}

Python

def code(x, y):
	return (0.5 / x) + (0.5 / y)

Julia

function code(x, y)
	return Float64(Float64(0.5 / x) + Float64(0.5 / y))
end

MATLAB

function tmp = code(x, y)
	tmp = (0.5 / x) + (0.5 / y);
end

Wolfram

code[x_, y_] := N[(N[(0.5 / x), $MachinePrecision] + N[(0.5 / y), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2023238 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, C"
  :precision binary64

  :herbie-target
  (+ (/ 0.5 x) (/ 0.5 y))

  (/ (+ x y) (* (* x 2.0) y)))