Linear.Projection:inversePerspective from linear-1.19.1.3, B

Percentage Accurate: 76.0% → 99.9%

Time: 2.7s

Alternatives: 7

Speedup: 0.6×

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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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: 76.0% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{0.5 + -0.5 \cdot \frac{y}{x}}{y}\\ \mathbf{if}\;x \leq -2 \cdot 10^{-8}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 470000000000:\\ \;\;\;\;\frac{0.5 \cdot \frac{x}{y} - 0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (+ 0.5 (* -0.5 (/ y x))) y)))
   (if (<= x -2e-8)
     t_0
     (if (<= x 470000000000.0) (/ (- (* 0.5 (/ x y)) 0.5) x) t_0))))

C

double code(double x, double y) {
	double t_0 = (0.5 + (-0.5 * (y / x))) / y;
	double tmp;
	if (x <= -2e-8) {
		tmp = t_0;
	} else if (x <= 470000000000.0) {
		tmp = ((0.5 * (x / y)) - 0.5) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (0.5d0 + ((-0.5d0) * (y / x))) / y
    if (x <= (-2d-8)) then
        tmp = t_0
    else if (x <= 470000000000.0d0) then
        tmp = ((0.5d0 * (x / y)) - 0.5d0) / x
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (0.5 + (-0.5 * (y / x))) / y;
	double tmp;
	if (x <= -2e-8) {
		tmp = t_0;
	} else if (x <= 470000000000.0) {
		tmp = ((0.5 * (x / y)) - 0.5) / x;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (0.5 + (-0.5 * (y / x))) / y
	tmp = 0
	if x <= -2e-8:
		tmp = t_0
	elif x <= 470000000000.0:
		tmp = ((0.5 * (x / y)) - 0.5) / x
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(0.5 + Float64(-0.5 * Float64(y / x))) / y)
	tmp = 0.0
	if (x <= -2e-8)
		tmp = t_0;
	elseif (x <= 470000000000.0)
		tmp = Float64(Float64(Float64(0.5 * Float64(x / y)) - 0.5) / x);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (0.5 + (-0.5 * (y / x))) / y;
	tmp = 0.0;
	if (x <= -2e-8)
		tmp = t_0;
	elseif (x <= 470000000000.0)
		tmp = ((0.5 * (x / y)) - 0.5) / x;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(0.5 + N[(-0.5 * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]}, If[LessEqual[x, -2e-8], t$95$0, If[LessEqual[x, 470000000000.0], N[(N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision], t$95$0]]]

Derivation

1. Split input into 2 regimes

2. if x < -2e-8 or 4.7e11 < x

   1. Initial program 76.0%

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

   2. Taylor expanded in y around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} + \frac{-1}{2} \cdot \frac{y}{x}}{y}} \]

   4. Applied rewrites 88.5%

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

   if -2e-8 < x < 4.7e11

   1. Initial program 76.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{x}{y} - \frac{1}{2}}{x}} \]

   4. Applied rewrites 88.4%

      \[\leadsto \color{blue}{\frac{0.5 \cdot \frac{x}{y} - 0.5}{x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 98.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{\left(x \cdot 2\right) \cdot y}\\ t_1 := \frac{x - y}{\left(y + y\right) \cdot x}\\ t_2 := \frac{0.5 \cdot \frac{x}{y} - 0.5}{x}\\ \mathbf{if}\;t\_0 \leq -\infty:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq -1 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{-45}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_0 \leq 4 \cdot 10^{+299}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (* (* x 2.0) y)))
        (t_1 (/ (- x y) (* (+ y y) x)))
        (t_2 (/ (- (* 0.5 (/ x y)) 0.5) x)))
   (if (<= t_0 (- INFINITY))
     t_2
     (if (<= t_0 -1e+59)
       t_1
       (if (<= t_0 4e-45) t_2 (if (<= t_0 4e+299) t_1 t_2))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / ((x * 2.0) * y);
	double t_1 = (x - y) / ((y + y) * x);
	double t_2 = ((0.5 * (x / y)) - 0.5) / x;
	double tmp;
	if (t_0 <= -((double) INFINITY)) {
		tmp = t_2;
	} else if (t_0 <= -1e+59) {
		tmp = t_1;
	} else if (t_0 <= 4e-45) {
		tmp = t_2;
	} else if (t_0 <= 4e+299) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / ((x * 2.0) * y);
	double t_1 = (x - y) / ((y + y) * x);
	double t_2 = ((0.5 * (x / y)) - 0.5) / x;
	double tmp;
	if (t_0 <= -Double.POSITIVE_INFINITY) {
		tmp = t_2;
	} else if (t_0 <= -1e+59) {
		tmp = t_1;
	} else if (t_0 <= 4e-45) {
		tmp = t_2;
	} else if (t_0 <= 4e+299) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / ((x * 2.0) * y)
	t_1 = (x - y) / ((y + y) * x)
	t_2 = ((0.5 * (x / y)) - 0.5) / x
	tmp = 0
	if t_0 <= -math.inf:
		tmp = t_2
	elif t_0 <= -1e+59:
		tmp = t_1
	elif t_0 <= 4e-45:
		tmp = t_2
	elif t_0 <= 4e+299:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(Float64(x * 2.0) * y))
	t_1 = Float64(Float64(x - y) / Float64(Float64(y + y) * x))
	t_2 = Float64(Float64(Float64(0.5 * Float64(x / y)) - 0.5) / x)
	tmp = 0.0
	if (t_0 <= Float64(-Inf))
		tmp = t_2;
	elseif (t_0 <= -1e+59)
		tmp = t_1;
	elseif (t_0 <= 4e-45)
		tmp = t_2;
	elseif (t_0 <= 4e+299)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / ((x * 2.0) * y);
	t_1 = (x - y) / ((y + y) * x);
	t_2 = ((0.5 * (x / y)) - 0.5) / x;
	tmp = 0.0;
	if (t_0 <= -Inf)
		tmp = t_2;
	elseif (t_0 <= -1e+59)
		tmp = t_1;
	elseif (t_0 <= 4e-45)
		tmp = t_2;
	elseif (t_0 <= 4e+299)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(N[(x * 2.0), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] / N[(N[(y + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(0.5 * N[(x / y), $MachinePrecision]), $MachinePrecision] - 0.5), $MachinePrecision] / x), $MachinePrecision]}, If[LessEqual[t$95$0, (-Infinity)], t$95$2, If[LessEqual[t$95$0, -1e+59], t$95$1, If[LessEqual[t$95$0, 4e-45], t$95$2, If[LessEqual[t$95$0, 4e+299], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -inf.0 or -9.99999999999999972e58 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 3.99999999999999994e-45 or 4.0000000000000002e299 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y))

   1. Initial program 76.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{x}{y} - \frac{1}{2}}{x}} \]

   4. Applied rewrites 88.4%

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

   if -inf.0 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < -9.99999999999999972e58 or 3.99999999999999994e-45 < (/.f64 (-.f64 x y) (*.f64 (*.f64 x #s(literal 2 binary64)) y)) < 4.0000000000000002e299

   1. Initial program 76.0%

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

   3. Applied rewrites 76.0%

      \[\leadsto \frac{x - y}{\color{blue}{\left(y + y\right) \cdot x}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 86.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{x - y}{\left(y + y\right) \cdot x}\\ \mathbf{if}\;x \leq -5.8 \cdot 10^{+148}:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq -2.15 \cdot 10^{-179}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \leq 3.7 \cdot 10^{-168}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{elif}\;x \leq 1.16 \cdot 10^{+69}:\\ \;\;\;\;t\_0\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (let* ((t_0 (/ (- x y) (* (+ y y) x))))
   (if (<= x -5.8e+148)
     (/ 0.5 y)
     (if (<= x -2.15e-179)
       t_0
       (if (<= x 3.7e-168) (/ -0.5 x) (if (<= x 1.16e+69) t_0 (/ 0.5 y)))))))

C

double code(double x, double y) {
	double t_0 = (x - y) / ((y + y) * x);
	double tmp;
	if (x <= -5.8e+148) {
		tmp = 0.5 / y;
	} else if (x <= -2.15e-179) {
		tmp = t_0;
	} else if (x <= 3.7e-168) {
		tmp = -0.5 / x;
	} else if (x <= 1.16e+69) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: t_0
    real(8) :: tmp
    t_0 = (x - y) / ((y + y) * x)
    if (x <= (-5.8d+148)) then
        tmp = 0.5d0 / y
    else if (x <= (-2.15d-179)) then
        tmp = t_0
    else if (x <= 3.7d-168) then
        tmp = (-0.5d0) / x
    else if (x <= 1.16d+69) then
        tmp = t_0
    else
        tmp = 0.5d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x - y) / ((y + y) * x);
	double tmp;
	if (x <= -5.8e+148) {
		tmp = 0.5 / y;
	} else if (x <= -2.15e-179) {
		tmp = t_0;
	} else if (x <= 3.7e-168) {
		tmp = -0.5 / x;
	} else if (x <= 1.16e+69) {
		tmp = t_0;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x - y) / ((y + y) * x)
	tmp = 0
	if x <= -5.8e+148:
		tmp = 0.5 / y
	elif x <= -2.15e-179:
		tmp = t_0
	elif x <= 3.7e-168:
		tmp = -0.5 / x
	elif x <= 1.16e+69:
		tmp = t_0
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x - y) / Float64(Float64(y + y) * x))
	tmp = 0.0
	if (x <= -5.8e+148)
		tmp = Float64(0.5 / y);
	elseif (x <= -2.15e-179)
		tmp = t_0;
	elseif (x <= 3.7e-168)
		tmp = Float64(-0.5 / x);
	elseif (x <= 1.16e+69)
		tmp = t_0;
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x - y) / ((y + y) * x);
	tmp = 0.0;
	if (x <= -5.8e+148)
		tmp = 0.5 / y;
	elseif (x <= -2.15e-179)
		tmp = t_0;
	elseif (x <= 3.7e-168)
		tmp = -0.5 / x;
	elseif (x <= 1.16e+69)
		tmp = t_0;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x - y), $MachinePrecision] / N[(N[(y + y), $MachinePrecision] * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -5.8e+148], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, -2.15e-179], t$95$0, If[LessEqual[x, 3.7e-168], N[(-0.5 / x), $MachinePrecision], If[LessEqual[x, 1.16e+69], t$95$0, N[(0.5 / y), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if x < -5.7999999999999999e148 or 1.16000000000000005e69 < x

   1. Initial program 76.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]

   4. Applied rewrites 50.2%

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

   if -5.7999999999999999e148 < x < -2.15000000000000013e-179 or 3.69999999999999997e-168 < x < 1.16000000000000005e69

   1. Initial program 76.0%

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

   3. Applied rewrites 76.0%

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

   if -2.15000000000000013e-179 < x < 3.69999999999999997e-168

   1. Initial program 76.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{x}} \]

   4. Applied rewrites 51.1%

      \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 4: 72.8% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -61000:\\ \;\;\;\;\frac{0.5}{y}\\ \mathbf{elif}\;x \leq 1.8 \cdot 10^{-110}:\\ \;\;\;\;\frac{-0.5}{x}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.5}{y}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y)
 :precision binary64
 (if (<= x -61000.0) (/ 0.5 y) (if (<= x 1.8e-110) (/ -0.5 x) (/ 0.5 y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -61000.0) {
		tmp = 0.5 / y;
	} else if (x <= 1.8e-110) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8) :: tmp
    if (x <= (-61000.0d0)) then
        tmp = 0.5d0 / y
    else if (x <= 1.8d-110) then
        tmp = (-0.5d0) / x
    else
        tmp = 0.5d0 / y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -61000.0) {
		tmp = 0.5 / y;
	} else if (x <= 1.8e-110) {
		tmp = -0.5 / x;
	} else {
		tmp = 0.5 / y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -61000.0:
		tmp = 0.5 / y
	elif x <= 1.8e-110:
		tmp = -0.5 / x
	else:
		tmp = 0.5 / y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -61000.0)
		tmp = Float64(0.5 / y);
	elseif (x <= 1.8e-110)
		tmp = Float64(-0.5 / x);
	else
		tmp = Float64(0.5 / y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -61000.0)
		tmp = 0.5 / y;
	elseif (x <= 1.8e-110)
		tmp = -0.5 / x;
	else
		tmp = 0.5 / y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -61000.0], N[(0.5 / y), $MachinePrecision], If[LessEqual[x, 1.8e-110], N[(-0.5 / x), $MachinePrecision], N[(0.5 / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -61000 or 1.79999999999999997e-110 < x

   1. Initial program 76.0%

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

   2. Taylor expanded in x around inf

      \[\leadsto \color{blue}{\frac{\frac{1}{2}}{y}} \]

   4. Applied rewrites 50.2%

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

   if -61000 < x < 1.79999999999999997e-110

   1. Initial program 76.0%

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

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{x}} \]

   4. Applied rewrites 51.1%

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

Alternative 5: 51.1% 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

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    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 76.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{x}} \]

4. Applied rewrites 51.1%

   \[\leadsto \color{blue}{\frac{-0.5}{x}} \]
5. Add Preprocessing

Alternative 6: 3.4% accurate, 3.3× speedup

Math

\[\begin{array}{l} \\ 2 \cdot 0.5 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 (* 2.0 0.5))

C

double code(double x, double y) {
	return 2.0 * 0.5;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0 * 0.5d0
end function

Java

public static double code(double x, double y) {
	return 2.0 * 0.5;
}

Python

def code(x, y):
	return 2.0 * 0.5

Julia

function code(x, y)
	return Float64(2.0 * 0.5)
end

MATLAB

function tmp = code(x, y)
	tmp = 2.0 * 0.5;
end

Wolfram

code[x_, y_] := N[(2.0 * 0.5), $MachinePrecision]

Derivation

1. Initial program 76.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{x}{y} - \frac{1}{2}}{x}} \]

4. Applied rewrites 88.4%

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

6. Applied rewrites 3.4%

   \[\leadsto 2 \cdot \color{blue}{0.5} \]
7. Add Preprocessing

Alternative 7: 3.4% accurate, 12.9× speedup

Math

\[\begin{array}{l} \\ 2 \end{array} \]

FPCore

(FPCore (x y) :precision binary64 2.0)

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = 2.0d0
end function

Java

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

Python

def code(x, y):
	return 2.0

Julia

function code(x, y)
	return 2.0
end

MATLAB

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

Wolfram

code[x_, y_] := 2.0

Derivation

1. Initial program 76.0%

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

2. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\frac{\frac{1}{2} \cdot \frac{x}{y} - \frac{1}{2}}{x}} \]

4. Applied rewrites 88.4%

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

6. Applied rewrites 3.4%

   \[\leadsto 2 \cdot \color{blue}{0.5} \]

8. Applied rewrites 3.4%

   \[\leadsto 2 \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2025159 
(FPCore (x y)
  :name "Linear.Projection:inversePerspective from linear-1.19.1.3, B"
  :precision binary64
  (/ (- x y) (* (* x 2.0) y)))