Diagrams.Solve.Polynomial:quartForm from diagrams-solve-0.1, D

Percentage Accurate: 100.0% → 100.0%

Time: 1.3s

Alternatives: 4

Speedup: 1.2×

Specification

Math

\[\frac{x \cdot y}{2} - \frac{z}{8} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * y) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	((x * y) / (2)) - (z / (8))
END code

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\frac{x \cdot y}{2} - \frac{z}{8} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (- (/ (* x y) 2.0) (/ z 8.0)))

C

double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = ((x * y) / 2.0d0) - (z / 8.0d0)
end function

Java

public static double code(double x, double y, double z) {
	return ((x * y) / 2.0) - (z / 8.0);
}

Python

def code(x, y, z):
	return ((x * y) / 2.0) - (z / 8.0)

Julia

function code(x, y, z)
	return Float64(Float64(Float64(x * y) / 2.0) - Float64(z / 8.0))
end

MATLAB

function tmp = code(x, y, z)
	tmp = ((x * y) / 2.0) - (z / 8.0);
end

Wolfram

code[x_, y_, z_] := N[(N[(N[(x * y), $MachinePrecision] / 2.0), $MachinePrecision] - N[(z / 8.0), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	((x * y) / (2)) - (z / (8))
END code

Alternative 1: 100.0% accurate, 1.2× speedup

Math

\[\mathsf{fma}\left(x, 0.5 \cdot y, -0.125 \cdot z\right) \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (fma x (* 0.5 y) (* -0.125 z)))

C

double code(double x, double y, double z) {
	return fma(x, (0.5 * y), (-0.125 * z));
}

Julia

function code(x, y, z)
	return fma(x, Float64(0.5 * y), Float64(-0.125 * z))
end

Wolfram

code[x_, y_, z_] := N[(x * N[(0.5 * y), $MachinePrecision] + N[(-0.125 * z), $MachinePrecision]), $MachinePrecision]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(x * ((5e-1) * y)) + ((-125e-3) * z)
END code

Derivation

1. Initial program 100.0%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]
2. Step-by-step derivation

3. Applied rewrites 100.0%

   \[\leadsto \mathsf{fma}\left(x, 0.5 \cdot y, -0.125 \cdot z\right) \]
4. Add Preprocessing

Alternative 2: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := x \cdot \left(0.5 \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -5 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot y \leq 5 \cdot 10^{-115}:\\ \;\;\;\;-0.125 \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* x (* 0.5 y))))
  (if (<= (* x y) -5e-11)
    t_0
    (if (<= (* x y) 5e-115) (* -0.125 z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = x * (0.5 * y);
	double tmp;
	if ((x * y) <= -5e-11) {
		tmp = t_0;
	} else if ((x * y) <= 5e-115) {
		tmp = -0.125 * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = x * (0.5d0 * y)
    if ((x * y) <= (-5d-11)) then
        tmp = t_0
    else if ((x * y) <= 5d-115) then
        tmp = (-0.125d0) * z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = x * (0.5 * y);
	double tmp;
	if ((x * y) <= -5e-11) {
		tmp = t_0;
	} else if ((x * y) <= 5e-115) {
		tmp = -0.125 * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = x * (0.5 * y)
	tmp = 0
	if (x * y) <= -5e-11:
		tmp = t_0
	elif (x * y) <= 5e-115:
		tmp = -0.125 * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(x * Float64(0.5 * y))
	tmp = 0.0
	if (Float64(x * y) <= -5e-11)
		tmp = t_0;
	elseif (Float64(x * y) <= 5e-115)
		tmp = Float64(-0.125 * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = x * (0.5 * y);
	tmp = 0.0;
	if ((x * y) <= -5e-11)
		tmp = t_0;
	elseif ((x * y) <= 5e-115)
		tmp = -0.125 * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(x * N[(0.5 * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5e-11], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 5e-115], N[(-0.125 * z), $MachinePrecision], t$95$0]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET t_0 = (x * ((5e-1) * y)) IN
		LET tmp_1 = IF ((x * y) <= (50000000000000002532245115846429047000311989752552678034498009382448470705408296023492374608185940085802109777202177840279277707015570576569178746282335047908242924101834562519900509625714227417248975325003782562360654965752209815932575397863348695633725959288671192332442305521211789454127938370220363140106201171875e-431)) THEN ((-125e-3) * z) ELSE t_0 ENDIF IN
		LET tmp = IF ((x * y) <= (-50000000000000001821609865774887078958277353279981980449520051479339599609375e-87)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.0000000000000002e-11 or 5.0000000000000003e-115 < (*.f64 x y)

   1. Initial program 100.0%

      \[\frac{x \cdot y}{2} - \frac{z}{8} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{8} \cdot z \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto -0.125 \cdot z \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot y\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 50.6%

      \[\leadsto 0.5 \cdot \left(x \cdot y\right) \]
   8. Step-by-step derivation

   9. Applied rewrites 50.6%

      \[\leadsto x \cdot \left(0.5 \cdot y\right) \]

   if -5.0000000000000002e-11 < (*.f64 x y) < 5.0000000000000003e-115

   1. Initial program 100.0%

      \[\frac{x \cdot y}{2} - \frac{z}{8} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{8} \cdot z \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto -0.125 \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 77.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_0 := 0.5 \cdot \left(x \cdot y\right)\\ \mathbf{if}\;x \cdot y \leq -5.373808364658808 \cdot 10^{-11}:\\ \;\;\;\;t\_0\\ \mathbf{elif}\;x \cdot y \leq 5.391364902004677 \cdot 10^{-103}:\\ \;\;\;\;-0.125 \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_0\\ \end{array} \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (let* ((t_0 (* 0.5 (* x y))))
  (if (<= (* x y) -5.373808364658808e-11)
    t_0
    (if (<= (* x y) 5.391364902004677e-103) (* -0.125 z) t_0))))

C

double code(double x, double y, double z) {
	double t_0 = 0.5 * (x * y);
	double tmp;
	if ((x * y) <= -5.373808364658808e-11) {
		tmp = t_0;
	} else if ((x * y) <= 5.391364902004677e-103) {
		tmp = -0.125 * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8) :: t_0
    real(8) :: tmp
    t_0 = 0.5d0 * (x * y)
    if ((x * y) <= (-5.373808364658808d-11)) then
        tmp = t_0
    else if ((x * y) <= 5.391364902004677d-103) then
        tmp = (-0.125d0) * z
    else
        tmp = t_0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z) {
	double t_0 = 0.5 * (x * y);
	double tmp;
	if ((x * y) <= -5.373808364658808e-11) {
		tmp = t_0;
	} else if ((x * y) <= 5.391364902004677e-103) {
		tmp = -0.125 * z;
	} else {
		tmp = t_0;
	}
	return tmp;
}

Python

def code(x, y, z):
	t_0 = 0.5 * (x * y)
	tmp = 0
	if (x * y) <= -5.373808364658808e-11:
		tmp = t_0
	elif (x * y) <= 5.391364902004677e-103:
		tmp = -0.125 * z
	else:
		tmp = t_0
	return tmp

Julia

function code(x, y, z)
	t_0 = Float64(0.5 * Float64(x * y))
	tmp = 0.0
	if (Float64(x * y) <= -5.373808364658808e-11)
		tmp = t_0;
	elseif (Float64(x * y) <= 5.391364902004677e-103)
		tmp = Float64(-0.125 * z);
	else
		tmp = t_0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z)
	t_0 = 0.5 * (x * y);
	tmp = 0.0;
	if ((x * y) <= -5.373808364658808e-11)
		tmp = t_0;
	elseif ((x * y) <= 5.391364902004677e-103)
		tmp = -0.125 * z;
	else
		tmp = t_0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_] := Block[{t$95$0 = N[(0.5 * N[(x * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x * y), $MachinePrecision], -5.373808364658808e-11], t$95$0, If[LessEqual[N[(x * y), $MachinePrecision], 5.391364902004677e-103], N[(-0.125 * z), $MachinePrecision], t$95$0]]]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	LET t_0 = ((5e-1) * (x * y)) IN
		LET tmp_1 = IF ((x * y) <= (53913649020046768688612367743278967712848165295837610976449159769879400796666202291688111631601870372965031932476203407054068749686774238906571940582653953875745923790332280249768674829315787278117869014480190487704842820876986990337901325477640876275564618680391504312865436077117919921875e-392)) THEN ((-125e-3) * z) ELSE t_0 ENDIF IN
		LET tmp = IF ((x * y) <= (-53738083646588077531584356898660434324888068857717371429316699504852294921875e-87)) THEN t_0 ELSE tmp_1 ENDIF IN
	tmp
END code

Derivation

1. Split input into 2 regimes

2. if (*.f64 x y) < -5.3738083646588078e-11 or 5.3913649020046769e-103 < (*.f64 x y)

   1. Initial program 100.0%

      \[\frac{x \cdot y}{2} - \frac{z}{8} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{8} \cdot z \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto -0.125 \cdot z \]

   5. Taylor expanded in x around inf

      \[\leadsto \frac{1}{2} \cdot \left(x \cdot y\right) \]
   6. Step-by-step derivation

   7. Applied rewrites 50.6%

      \[\leadsto 0.5 \cdot \left(x \cdot y\right) \]

   if -5.3738083646588078e-11 < (*.f64 x y) < 5.3913649020046769e-103

   1. Initial program 100.0%

      \[\frac{x \cdot y}{2} - \frac{z}{8} \]

   2. Taylor expanded in x around 0

      \[\leadsto \frac{-1}{8} \cdot z \]
   3. Step-by-step derivation

   4. Applied rewrites 51.3%

      \[\leadsto -0.125 \cdot z \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 51.3% accurate, 3.5× speedup

Math

\[-0.125 \cdot z \]

FPCore

(FPCore (x y z)
  :precision binary64
  :pre TRUE
  (* -0.125 z))

C

double code(double x, double y, double z) {
	return -0.125 * z;
}

Fortran

real(8) function code(x, y, z)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    code = (-0.125d0) * z
end function

Java

public static double code(double x, double y, double z) {
	return -0.125 * z;
}

Python

def code(x, y, z):
	return -0.125 * z

Julia

function code(x, y, z)
	return Float64(-0.125 * z)
end

MATLAB

function tmp = code(x, y, z)
	tmp = -0.125 * z;
end

Wolfram

code[x_, y_, z_] := N[(-0.125 * z), $MachinePrecision]

ReFlow

f(x, y, z):
	x in [-inf, +inf],
	y in [-inf, +inf],
	z in [-inf, +inf]
code: THEORY
BEGIN
f(x, y, z: real): real =
	(-125e-3) * z
END code

Derivation

1. Initial program 100.0%

   \[\frac{x \cdot y}{2} - \frac{z}{8} \]

2. Taylor expanded in x around 0

   \[\leadsto \frac{-1}{8} \cdot z \]
3. Step-by-step derivation

4. Applied rewrites 51.3%

   \[\leadsto -0.125 \cdot z \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2026092 
(FPCore (x y z)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, D"
  :precision binary64
  (- (/ (* x y) 2.0) (/ z 8.0)))