Linear.Projection:perspective from linear-1.19.1.3, B

Percentage Accurate: 77.3% → 99.9%

Time: 1.7s

Alternatives: 5

Speedup: 0.7×

Specification

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ (* (* x 2.0) y) (- x y)))

C

double code(double x, double y) {
	return ((x * 2.0) * 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 * 2.0d0) * y) / (x - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * 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 * (2)) * y) / (x - y)
END code

Initial Program: 77.3% accurate, 1.0× speedup

Math

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

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (/ (* (* x 2.0) y) (- x y)))

C

double code(double x, double y) {
	return ((x * 2.0) * 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 * 2.0d0) * y) / (x - y)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_] := N[(N[(N[(x * 2.0), $MachinePrecision] * 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 * (2)) * y) / (x - y)
END code

Alternative 1: 99.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -13996.61964560276:\\ \;\;\;\;\left(y + y\right) \cdot \frac{x}{x - y}\\ \mathbf{elif}\;x \leq 1.6822977259161094 \cdot 10^{-10}:\\ \;\;\;\;\left(x + x\right) \cdot \frac{y}{x - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{0.5 - \frac{y}{x + x}}\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= x -13996.61964560276)
  (* (+ y y) (/ x (- x y)))
  (if (<= x 1.6822977259161094e-10)
    (* (+ x x) (/ y (- x y)))
    (/ y (- 0.5 (/ y (+ x x)))))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -13996.61964560276) {
		tmp = (y + y) * (x / (x - y));
	} else if (x <= 1.6822977259161094e-10) {
		tmp = (x + x) * (y / (x - y));
	} else {
		tmp = y / (0.5 - (y / (x + x)));
	}
	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 <= (-13996.61964560276d0)) then
        tmp = (y + y) * (x / (x - y))
    else if (x <= 1.6822977259161094d-10) then
        tmp = (x + x) * (y / (x - y))
    else
        tmp = y / (0.5d0 - (y / (x + x)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -13996.61964560276) {
		tmp = (y + y) * (x / (x - y));
	} else if (x <= 1.6822977259161094e-10) {
		tmp = (x + x) * (y / (x - y));
	} else {
		tmp = y / (0.5 - (y / (x + x)));
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -13996.61964560276:
		tmp = (y + y) * (x / (x - y))
	elif x <= 1.6822977259161094e-10:
		tmp = (x + x) * (y / (x - y))
	else:
		tmp = y / (0.5 - (y / (x + x)))
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -13996.61964560276)
		tmp = Float64(Float64(y + y) * Float64(x / Float64(x - y)));
	elseif (x <= 1.6822977259161094e-10)
		tmp = Float64(Float64(x + x) * Float64(y / Float64(x - y)));
	else
		tmp = Float64(y / Float64(0.5 - Float64(y / Float64(x + x))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -13996.61964560276)
		tmp = (y + y) * (x / (x - y));
	elseif (x <= 1.6822977259161094e-10)
		tmp = (x + x) * (y / (x - y));
	else
		tmp = y / (0.5 - (y / (x + x)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -13996.61964560276], N[(N[(y + y), $MachinePrecision] * N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.6822977259161094e-10], N[(N[(x + x), $MachinePrecision] * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(0.5 - N[(y / N[(x + x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp_1 = IF (x <= (168229772591610942516408030476620337478887989846043637953698635101318359375e-84)) THEN ((x + x) * (y / (x - y))) ELSE (y / ((5e-1) - (y / (x + x)))) ENDIF IN
	LET tmp = IF (x <= (-13996619645602759192115627229213714599609375e-39)) THEN ((y + y) * (x / (x - y))) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 3 regimes

2. if x < -13996.619645602759

   1. Initial program 77.3%

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

   3. Applied rewrites 88.2%

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

   if -13996.619645602759 < x < 1.6822977259161094e-10

   1. Initial program 77.3%

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

   3. Applied rewrites 89.0%

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

   if 1.6822977259161094e-10 < x

   1. Initial program 77.3%

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

   3. Applied rewrites 77.1%

      \[\leadsto \frac{1}{\frac{x - y}{y \cdot \left(x + x\right)}} \]
   4. Step-by-step derivation

   5. Applied rewrites 87.5%

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

   7. Applied rewrites 87.6%

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

Alternative 2: 99.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(x + x\right) \cdot \frac{y}{x - y}\\ \mathbf{if}\;y \leq -1.712317240886347 \cdot 10^{+24}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 4.528461901083382 \cdot 10^{+77}:\\ \;\;\;\;\left(y + y\right) \cdot \frac{x}{x - y}\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* (+ x x) (/ y (- x y)))))
  (if (<= y -1.712317240886347e+24)
    t_0
    (if (<= y 4.528461901083382e+77) (* (+ y y) (/ x (- x y))) t_0))))

C

double code(double x, double y) {
	double t_0 = (x + x) * (y / (x - y));
	double tmp;
	if (y <= -1.712317240886347e+24) {
		tmp = t_0;
	} else if (y <= 4.528461901083382e+77) {
		tmp = (y + y) * (x / (x - y));
	} else {
		tmp = t_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) :: t_0
    real(8) :: tmp
    t_0 = (x + x) * (y / (x - y))
    if (y <= (-1.712317240886347d+24)) then
        tmp = t_0
    else if (y <= 4.528461901083382d+77) then
        tmp = (y + y) * (x / (x - y))
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double t_0 = (x + x) * (y / (x - y));
	double tmp;
	if (y <= -1.712317240886347e+24) {
		tmp = t_0;
	} else if (y <= 4.528461901083382e+77) {
		tmp = (y + y) * (x / (x - y));
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y):
	t_0 = (x + x) * (y / (x - y))
	tmp = 0
	if y <= -1.712317240886347e+24:
		tmp = t_0
	elif y <= 4.528461901083382e+77:
		tmp = (y + y) * (x / (x - y))
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y)
	t_0 = Float64(Float64(x + x) * Float64(y / Float64(x - y)))
	tmp = 0.0
	if (y <= -1.712317240886347e+24)
		tmp = t_0;
	elseif (y <= 4.528461901083382e+77)
		tmp = Float64(Float64(y + y) * Float64(x / Float64(x - y)));
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	t_0 = (x + x) * (y / (x - y));
	tmp = 0.0;
	if (y <= -1.712317240886347e+24)
		tmp = t_0;
	elseif (y <= 4.528461901083382e+77)
		tmp = (y + y) * (x / (x - y));
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + x), $MachinePrecision] * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.712317240886347e+24], t$95$0, If[LessEqual[y, 4.528461901083382e+77], N[(N[(y + y), $MachinePrecision] * N[(x / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$0]]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET t_0 = ((x + x) * (y / (x - y))) IN
		LET tmp_1 = IF (y <= (452846190108338205639346410490344069348877864692958048651167311389807469395968)) THEN ((y + y) * (x / (x - y))) ELSE t_0 ENDIF IN
		LET tmp = IF (y <= (-1712317240886347050254336)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < -1.7123172408863471e24 or 4.5284619010833821e77 < y

   1. Initial program 77.3%

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

   3. Applied rewrites 89.0%

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

   if -1.7123172408863471e24 < y < 4.5284619010833821e77

   1. Initial program 77.3%

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

   3. Applied rewrites 88.2%

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

Alternative 3: 93.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := \left(x + x\right) \cdot \frac{y}{x - y}\\ \mathbf{if}\;y \leq -1.1475811057250325 \cdot 10^{-242}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;y \leq 3.3003929900051086 \cdot 10^{-162}:\\ \;\;\;\;2 \cdot \mathsf{fma}\left(y, \frac{y}{x}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* (+ x x) (/ y (- x y)))))
  (if (<= y -1.1475811057250325e-242)
    t_0
    (if (<= y 3.3003929900051086e-162)
      (* 2.0 (fma y (/ y x) y))
      t_0))))

C

double code(double x, double y) {
	double t_0 = (x + x) * (y / (x - y));
	double tmp;
	if (y <= -1.1475811057250325e-242) {
		tmp = t_0;
	} else if (y <= 3.3003929900051086e-162) {
		tmp = 2.0 * fma(y, (y / x), y);
	} else {
		tmp = t_0;
	}
	return tmp;
}

Julia

function code(x, y)
	t_0 = Float64(Float64(x + x) * Float64(y / Float64(x - y)))
	tmp = 0.0
	if (y <= -1.1475811057250325e-242)
		tmp = t_0;
	elseif (y <= 3.3003929900051086e-162)
		tmp = Float64(2.0 * fma(y, Float64(y / x), y));
	else
		tmp = t_0;
	end
	return tmp
end

Wolfram

code[x_, y_] := Block[{t$95$0 = N[(N[(x + x), $MachinePrecision] * N[(y / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.1475811057250325e-242], t$95$0, If[LessEqual[y, 3.3003929900051086e-162], N[(2.0 * N[(y * N[(y / x), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision], t$95$0]]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET t_0 = ((x + x) * (y / (x - y))) IN
		LET tmp_1 = IF (y <= (33003929900051086347633598244156977954691151604997871907467177159832261398428432006018386061616764705292641951582290021070506605369882130874562431347736456645566339031447937487061916350476490439019324430673195347128433523832131311258936944110662937357325319337185152867380509671240634025926589954312950667530949858050817133788838994739852497957907901976772319859700447651883054226546430898192596714579849503934383392333984375e-586)) THEN ((2) * ((y * (y / x)) + y)) ELSE t_0 ENDIF IN
		LET tmp = IF (y <= (-114758110572503251141730386607988073827233731336094270904548315016248603744169163772647436088509301026416197194398379093526382737549206915285146415828362700843542498130083997403840439276482232652111888735691617304148039009900291961400231508412249614303588035984396211956383739474470627363590810685403534313467749475416057269045303588270493753328734396097084524629770525191062035924065657671862236091119307639050151711710238199316014633732709504747714518442602818925504881832045587457928815500522681107907329061881305551446435885618696033397517038479632377185429558619981886546934646275985869579017162322998046875e-853)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if y < -1.1475811057250325e-242 or 3.3003929900051086e-162 < y

   1. Initial program 77.3%

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

   3. Applied rewrites 89.0%

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

   if -1.1475811057250325e-242 < y < 3.3003929900051086e-162

   1. Initial program 77.3%

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

   2. Taylor expanded in y around 0

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

   4. Applied rewrites 50.1%

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

   6. Applied rewrites 50.1%

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

Alternative 4: 74.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \mathbf{if}\;x \leq -5.039907711792451 \cdot 10^{-56}:\\ \;\;\;\;y + y\\ \mathbf{elif}\;x \leq 4.1876733463546205 \cdot 10^{+30}:\\ \;\;\;\;-2 \cdot x\\ \mathbf{else}:\\ \;\;\;\;y + y\\ \end{array} \]

FPCore

(FPCore (x y)
  :precision binary64
  :pre TRUE
  (if (<= x -5.039907711792451e-56)
  (+ y y)
  (if (<= x 4.1876733463546205e+30) (* -2.0 x) (+ y y))))

C

double code(double x, double y) {
	double tmp;
	if (x <= -5.039907711792451e-56) {
		tmp = y + y;
	} else if (x <= 4.1876733463546205e+30) {
		tmp = -2.0 * x;
	} else {
		tmp = y + y;
	}
	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 <= (-5.039907711792451d-56)) then
        tmp = y + y
    else if (x <= 4.1876733463546205d+30) then
        tmp = (-2.0d0) * x
    else
        tmp = y + y
    end if
    code = tmp
end function

Java

public static double code(double x, double y) {
	double tmp;
	if (x <= -5.039907711792451e-56) {
		tmp = y + y;
	} else if (x <= 4.1876733463546205e+30) {
		tmp = -2.0 * x;
	} else {
		tmp = y + y;
	}
	return tmp;
}

Python

def code(x, y):
	tmp = 0
	if x <= -5.039907711792451e-56:
		tmp = y + y
	elif x <= 4.1876733463546205e+30:
		tmp = -2.0 * x
	else:
		tmp = y + y
	return tmp

Julia

function code(x, y)
	tmp = 0.0
	if (x <= -5.039907711792451e-56)
		tmp = Float64(y + y);
	elseif (x <= 4.1876733463546205e+30)
		tmp = Float64(-2.0 * x);
	else
		tmp = Float64(y + y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y)
	tmp = 0.0;
	if (x <= -5.039907711792451e-56)
		tmp = y + y;
	elseif (x <= 4.1876733463546205e+30)
		tmp = -2.0 * x;
	else
		tmp = y + y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_] := If[LessEqual[x, -5.039907711792451e-56], N[(y + y), $MachinePrecision], If[LessEqual[x, 4.1876733463546205e+30], N[(-2.0 * x), $MachinePrecision], N[(y + y), $MachinePrecision]]]

ReFlow

f(x, y):
	x in [-inf, +inf],
	y in [-inf, +inf]
code: THEORY
BEGIN
f(x, y: real): real =
	LET tmp_1 = IF (x <= (4187673346354620473416423047168)) THEN ((-2) * x) ELSE (y + y) ENDIF IN
	LET tmp = IF (x <= (-503990771179245087994483158180417636751518472116048142633389048672491776078072589798927984249697471122399245012078916984904270611085258329241154395816693067899905145168304443359375e-235)) THEN (y + y) ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if x < -5.0399077117924509e-56 or 4.1876733463546205e30 < x

   1. Initial program 77.3%

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

   2. Taylor expanded in x around inf

      \[\leadsto 2 \cdot y \]
   3. Step-by-step derivation

   4. Applied rewrites 50.5%

      \[\leadsto 2 \cdot y \]
   5. Step-by-step derivation

   6. Applied rewrites 50.5%

      \[\leadsto y + y \]

   if -5.0399077117924509e-56 < x < 4.1876733463546205e30

   1. Initial program 77.3%

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

   2. Taylor expanded in x around 0

      \[\leadsto -2 \cdot x \]
   3. Step-by-step derivation

   4. Applied rewrites 50.9%

      \[\leadsto -2 \cdot x \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 50.5% accurate, 3.7× speedup

Math

\[y + y \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

ReFlow

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

Derivation

1. Initial program 77.3%

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

2. Taylor expanded in x around inf

   \[\leadsto 2 \cdot y \]
3. Step-by-step derivation

4. Applied rewrites 50.5%

   \[\leadsto 2 \cdot y \]
5. Step-by-step derivation

6. Applied rewrites 50.5%

   \[\leadsto y + y \]
7. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y)
  :name "Linear.Projection:perspective from linear-1.19.1.3, B"
  :precision binary64
  (/ (* (* x 2.0) y) (- x y)))