Linear.Projection:perspective from linear-1.19.1.3, A

Percentage Accurate: 100.0% → 100.0%

Time: 1.2s

Alternatives: 7

Speedup: 1.0×

Specification

Math

\[\frac{x + y}{x - y} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ (+ x y) (- x y)))

C

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

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / (x - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(x + y) / (x - y)
END code

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\frac{x + y}{x - y} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ (+ x y) (- x y)))

C

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

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = (x + y) / (x - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	(x + y) / (x - y)
END code

Alternative 1: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{x - y} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, -2, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{x}, 2, 1\right)\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (/ (+ x y) (- x y)) -0.5)
  (fma (/ x y) -2.0 -1.0)
  (fma (/ y x) 2.0 1.0)))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = fma((x / y), -2.0, -1.0);
	} else {
		tmp = fma((y / x), 2.0, 1.0);
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) / Float64(x - y)) <= -0.5)
		tmp = fma(Float64(x / y), -2.0, -1.0);
	else
		tmp = fma(Float64(y / x), 2.0, 1.0);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(x / y), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision], N[(N[(y / x), $MachinePrecision] * 2.0 + 1.0), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((x + y) / (x - y)) <= (-5e-1)) THEN (((x / y) * (-2)) + (-1)) ELSE (((y / x) * (2)) + (1)) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < -0.5

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0

      \[\leadsto -2 \cdot \frac{x}{y} - 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 51.1%

      \[\leadsto -2 \cdot \frac{x}{y} - 1 \]
   5. Step-by-step derivation

   6. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, -2, -1\right) \]

   if -0.5 < (/.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in y around 0

      \[\leadsto 1 + 2 \cdot \frac{y}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.9%

      \[\leadsto 1 + 2 \cdot \frac{y}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.9%

      \[\leadsto \mathsf{fma}\left(\frac{y}{x}, 2, 1\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 2: 99.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{x - y} \leq -0.5:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{y}, -2, -1\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, y, x\right)}{x}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (/ (+ x y) (- x y)) -0.5)
  (fma (/ x y) -2.0 -1.0)
  (/ (fma 2.0 y x) x)))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = fma((x / y), -2.0, -1.0);
	} else {
		tmp = fma(2.0, y, x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) / Float64(x - y)) <= -0.5)
		tmp = fma(Float64(x / y), -2.0, -1.0);
	else
		tmp = Float64(fma(2.0, y, x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], -0.5], N[(N[(x / y), $MachinePrecision] * -2.0 + -1.0), $MachinePrecision], N[(N[(2.0 * y + x), $MachinePrecision] / x), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((x + y) / (x - y)) <= (-5e-1)) THEN (((x / y) * (-2)) + (-1)) ELSE ((((2) * y) + x) / x) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < -0.5

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0

      \[\leadsto -2 \cdot \frac{x}{y} - 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 51.1%

      \[\leadsto -2 \cdot \frac{x}{y} - 1 \]
   5. Step-by-step derivation

   6. Applied rewrites 51.1%

      \[\leadsto \mathsf{fma}\left(\frac{x}{y}, -2, -1\right) \]

   if -0.5 < (/.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in y around 0

      \[\leadsto 1 + 2 \cdot \frac{y}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.9%

      \[\leadsto 1 + 2 \cdot \frac{y}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.9%

      \[\leadsto \frac{\mathsf{fma}\left(2, y, x\right)}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{x - y} \leq -0.5:\\ \;\;\;\;\frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, y, x\right)}{x}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (/ (+ x y) (- x y)) -0.5) (/ y (- x y)) (/ (fma 2.0 y x) x)))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = y / (x - y);
	} else {
		tmp = fma(2.0, y, x) / x;
	}
	return tmp;
}

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) / Float64(x - y)) <= -0.5)
		tmp = Float64(y / Float64(x - y));
	else
		tmp = Float64(fma(2.0, y, x) / x);
	end
	return tmp
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], -0.5], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision], N[(N[(2.0 * y + x), $MachinePrecision] / x), $MachinePrecision]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((x + y) / (x - y)) <= (-5e-1)) THEN (y / (x - y)) ELSE ((((2) * y) + x) / x) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < -0.5

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x - y} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.1%

      \[\leadsto \frac{y}{x - y} \]

   if -0.5 < (/.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in y around 0

      \[\leadsto 1 + 2 \cdot \frac{y}{x} \]
   3. Step-by-step derivation

   4. Applied rewrites 50.9%

      \[\leadsto 1 + 2 \cdot \frac{y}{x} \]
   5. Step-by-step derivation

   6. Applied rewrites 50.9%

      \[\leadsto \frac{\mathsf{fma}\left(2, y, x\right)}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 98.0% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (/ (+ x y) (- x y)) -0.5) (/ y (- x y)) 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = y / (x - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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 + y) / (x - y)) <= (-0.5d0)) then
        tmp = y / (x - y)
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= -0.5) {
		tmp = y / (x - y);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) / (x - y)) <= -0.5:
		tmp = y / (x - y)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) / Float64(x - y)) <= -0.5)
		tmp = Float64(y / Float64(x - y));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) / (x - y)) <= -0.5)
		tmp = y / (x - y);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], -0.5], N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision], 1.0]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((x + y) / (x - y)) <= (-5e-1)) THEN (y / (x - y)) ELSE (1) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < -0.5

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{y}{x - y} \]
   3. Step-by-step derivation

   4. Applied rewrites 51.1%

      \[\leadsto \frac{y}{x - y} \]

   if -0.5 < (/.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around inf

      \[\leadsto 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 49.6%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 98.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x + y}{x - y} \leq 4.731472837266996 \cdot 10^{-222}:\\ \;\;\;\;-1\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= (/ (+ x y) (- x y)) 4.731472837266996e-222) -1.0 1.0))

C

double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= 4.731472837266996e-222) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

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 + y) / (x - y)) <= 4.731472837266996d-222) then
        tmp = -1.0d0
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (((x + y) / (x - y)) <= 4.731472837266996e-222) {
		tmp = -1.0;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if ((x + y) / (x - y)) <= 4.731472837266996e-222:
		tmp = -1.0
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (Float64(Float64(x + y) / Float64(x - y)) <= 4.731472837266996e-222)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (((x + y) / (x - y)) <= 4.731472837266996e-222)
		tmp = -1.0;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[N[(N[(x + y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision], 4.731472837266996e-222], -1.0, 1.0]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp = IF (((x + y) / (x - y)) <= (473147283726699585065293851524435128190922016333699055671791504033202925272543750520206277009555675015528260548249336085724343511072245950950391626280123116962803550854711619391945100915530189136478332173189165768498201566491010149983509988265524528973801113689207603874660658232879186955197282318134343113697197728453108173460468511653471980995767230165508145513483154768723497648772791019757386434616993332037049089313993140657421500438219506954795786635255153171776586993997967592115062145957106845622535911412997419958777721438192287450874573551118373870849609375e-788)) THEN (-1) ELSE (1) ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x y) (-.f64 x y)) < 4.7314728372669959e-222

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around 0

      \[\leadsto -1 \]
   3. Step-by-step derivation

   4. Applied rewrites 50.0%

      \[\leadsto -1 \]

   if 4.7314728372669959e-222 < (/.f64 (+.f64 x y) (-.f64 x y))

   1. Initial program 100.0%

      \[\frac{x + y}{x - y} \]

   2. Taylor expanded in x around inf

      \[\leadsto 1 \]
   3. Step-by-step derivation

   4. Applied rewrites 49.6%

      \[\leadsto 1 \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 50.0% accurate, 10.0× speedup

Math

\[-1 \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  -1.0)

C

double code(double x, double y) {
	return -1.0;
}

Fortran

real(8) function code(x, y)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    code = -1.0d0
end function

Java

public static double code(double x, double y) {
	return -1.0;
}

Python

def code(x, y):
	return -1.0

Julia

function code(x, y)
	return -1.0
end

MATLAB

function tmp = code(x, y)
	tmp = -1.0;
end

Wolfram

code[x_, y_] := -1.0

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	-1
END code

Derivation

1. Initial program 100.0%

   \[\frac{x + y}{x - y} \]

2. Taylor expanded in x around 0

   \[\leadsto -1 \]
3. Step-by-step derivation

4. Applied rewrites 50.0%

   \[\leadsto -1 \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, A"
  :precision binary64
  (/ (+ x y) (- x y)))