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

Percentage Accurate: 100.0% → 100.0%

Time: 4.0s

Alternatives: 5

Speedup: 1.0×

Specification

Math

\[\begin{array}{l} \\ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Python

def code(x, y, z, t):
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Initial Program: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))

C

double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (((1.0d0 / 8.0d0) * x) - ((y * z) / 2.0d0)) + t
end function

Java

public static double code(double x, double y, double z, double t) {
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
}

Python

def code(x, y, z, t):
	return (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t

Julia

function code(x, y, z, t)
	return Float64(Float64(Float64(Float64(1.0 / 8.0) * x) - Float64(Float64(y * z) / 2.0)) + t)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (((1.0 / 8.0) * x) - ((y * z) / 2.0)) + t;
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 / 8.0), $MachinePrecision] * x), $MachinePrecision] - N[(N[(y * z), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]

Alternative 1: 100.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ t + \left(x \cdot \frac{1}{8} - \frac{z \cdot y}{2}\right) \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ t (- (* x (/ 1.0 8.0)) (/ (* z y) 2.0))))

C

double code(double x, double y, double z, double t) {
	return t + ((x * (1.0 / 8.0)) - ((z * y) / 2.0));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t + ((x * (1.0d0 / 8.0d0)) - ((z * y) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return t + ((x * (1.0 / 8.0)) - ((z * y) / 2.0));
}

Python

def code(x, y, z, t):
	return t + ((x * (1.0 / 8.0)) - ((z * y) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(t + Float64(Float64(x * Float64(1.0 / 8.0)) - Float64(Float64(z * y) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t + ((x * (1.0 / 8.0)) - ((z * y) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(t + N[(N[(x * N[(1.0 / 8.0), $MachinePrecision]), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / 2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Add Preprocessing

3. Final simplification 100.0%

   \[\leadsto t + \left(x \cdot \frac{1}{8} - \frac{z \cdot y}{2}\right) \]
4. Add Preprocessing

Alternative 2: 88.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \mathbf{if}\;z \cdot y \leq -0.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot y \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (* -0.5 z) y t)))
   (if (<= (* z y) -0.2) t_1 (if (<= (* z y) 2e+54) (fma x 0.125 t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((-0.5 * z), y, t);
	double tmp;
	if ((z * y) <= -0.2) {
		tmp = t_1;
	} else if ((z * y) <= 2e+54) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(-0.5 * z), y, t)
	tmp = 0.0
	if (Float64(z * y) <= -0.2)
		tmp = t_1;
	elseif (Float64(z * y) <= 2e+54)
		tmp = fma(x, 0.125, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision]}, If[LessEqual[N[(z * y), $MachinePrecision], -0.2], t$95$1, If[LessEqual[N[(z * y), $MachinePrecision], 2e+54], N[(x * 0.125 + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -0.20000000000000001 or 2.0000000000000002e54 < (*.f64 y z)

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

      \[\leadsto \color{blue}{t - \frac{1}{2} \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{t + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(y \cdot z\right)} \]

      2. metadata-eval N/A

         \[\leadsto t + \color{blue}{\frac{-1}{2}} \cdot \left(y \cdot z\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(y \cdot z\right) + t} \]

      4. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(z \cdot y\right)} + t \]

      5. associate-*r* N/A

         \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot z\right) \cdot y} + t \]

      6. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot z, y, t\right)} \]

      7. lower-*.f64 91.2

         \[\leadsto \mathsf{fma}\left(\color{blue}{-0.5 \cdot z}, y, t\right) \]

   5. Applied rewrites 91.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot z, y, t\right)} \]

   if -0.20000000000000001 < (*.f64 y z) < 2.0000000000000002e54

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{8} \cdot x + t} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{8}} + t \]

      3. lower-fma.f64 91.3

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.125, t\right)} \]

   5. Applied rewrites 91.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.125, t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 91.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -0.2:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \mathbf{elif}\;z \cdot y \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 82.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -0.5 \cdot \left(z \cdot y\right)\\ \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+179}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot y \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* -0.5 (* z y))))
   (if (<= (* z y) -1e+179) t_1 (if (<= (* z y) 2e+54) (fma x 0.125 t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.5 * (z * y);
	double tmp;
	if ((z * y) <= -1e+179) {
		tmp = t_1;
	} else if ((z * y) <= 2e+54) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(-0.5 * Float64(z * y))
	tmp = 0.0
	if (Float64(z * y) <= -1e+179)
		tmp = t_1;
	elseif (Float64(z * y) <= 2e+54)
		tmp = fma(x, 0.125, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * y), $MachinePrecision], -1e+179], t$95$1, If[LessEqual[N[(z * y), $MachinePrecision], 2e+54], N[(x * 0.125 + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -9.9999999999999998e178 or 2.0000000000000002e54 < (*.f64 y z)

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(y \cdot z\right)} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(y \cdot z\right)} \]

      2. *-commutative N/A

         \[\leadsto \frac{-1}{2} \cdot \color{blue}{\left(z \cdot y\right)} \]

      3. lower-*.f64 85.3

         \[\leadsto -0.5 \cdot \color{blue}{\left(z \cdot y\right)} \]

   5. Applied rewrites 85.3%

      \[\leadsto \color{blue}{-0.5 \cdot \left(z \cdot y\right)} \]

   if -9.9999999999999998e178 < (*.f64 y z) < 2.0000000000000002e54

   1. Initial program 100.0%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
   4. Step-by-step derivation

      1. +-commutative N/A

         \[\leadsto \color{blue}{\frac{1}{8} \cdot x + t} \]

      2. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{8}} + t \]

      3. lower-fma.f64 86.6

         \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.125, t\right)} \]

   5. Applied rewrites 86.6%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.125, t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+179}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;z \cdot y \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 63.9% accurate, 5.6× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(x, 0.125, t\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (fma x 0.125 t))

C

double code(double x, double y, double z, double t) {
	return fma(x, 0.125, t);
}

Julia

function code(x, y, z, t)
	return fma(x, 0.125, t)
end

Wolfram

code[x_, y_, z_, t_] := N[(x * 0.125 + t), $MachinePrecision]

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Add Preprocessing

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{t + \frac{1}{8} \cdot x} \]
4. Step-by-step derivation

   1. +-commutative N/A

      \[\leadsto \color{blue}{\frac{1}{8} \cdot x + t} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{8}} + t \]

   3. lower-fma.f64 61.5

      \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.125, t\right)} \]

5. Applied rewrites 61.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(x, 0.125, t\right)} \]
6. Add Preprocessing

Alternative 5: 33.2% accurate, 6.5× speedup

Math

\[\begin{array}{l} \\ 0.125 \cdot x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* 0.125 x))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
2. Add Preprocessing

3. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{8} \cdot x} \]
4. Step-by-step derivation

   1. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{8}} \]

   2. lower-*.f64 28.3

      \[\leadsto \color{blue}{x \cdot 0.125} \]

5. Applied rewrites 28.3%

   \[\leadsto \color{blue}{x \cdot 0.125} \]

6. Final simplification 28.3%

   \[\leadsto 0.125 \cdot x \]
7. Add Preprocessing

Developer Target 1: 100.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \left(\frac{x}{8} + t\right) - \frac{z}{2} \cdot y \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- (+ (/ x 8.0) t) (* (/ z 2.0) y)))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024255 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (- (+ (/ x 8) t) (* (/ z 2) y)))

  (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))