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

Percentage Accurate: 100.0% → 100.0%

Time: 4.1s

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

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

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]

Derivation

1. Initial program 100.0%

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

Alternative 2: 87.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, z \cdot -0.5, t\right)\\ \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot z \leq 2000000000:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma y (* z -0.5) t)))
   (if (<= (* y z) -2e+29)
     t_1
     (if (<= (* y z) 2000000000.0) (fma 0.125 x t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(y, (z * -0.5), t);
	double tmp;
	if ((y * z) <= -2e+29) {
		tmp = t_1;
	} else if ((y * z) <= 2000000000.0) {
		tmp = fma(0.125, x, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(y, Float64(z * -0.5), t)
	tmp = 0.0
	if (Float64(y * z) <= -2e+29)
		tmp = t_1;
	elseif (Float64(y * z) <= 2000000000.0)
		tmp = fma(0.125, x, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z * -0.5), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -2e+29], t$95$1, If[LessEqual[N[(y * z), $MachinePrecision], 2000000000.0], N[(0.125 * x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.99999999999999983e29 or 2e9 < (*.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

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-*.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if -1.99999999999999983e29 < (*.f64 y z) < 2e9

   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 y 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. lower-fma.f64 91.5

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

   5. Applied rewrites 91.5%

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

Alternative 3: 83.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z \cdot -0.5\right)\\ \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot z \leq 10^{+130}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* z -0.5))))
   (if (<= (* y z) -2e+55) t_1 (if (<= (* y z) 1e+130) (fma 0.125 x t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (z * -0.5);
	double tmp;
	if ((y * z) <= -2e+55) {
		tmp = t_1;
	} else if ((y * z) <= 1e+130) {
		tmp = fma(0.125, x, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(z * -0.5))
	tmp = 0.0
	if (Float64(y * z) <= -2e+55)
		tmp = t_1;
	elseif (Float64(y * z) <= 1e+130)
		tmp = fma(0.125, x, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -2e+55], t$95$1, If[LessEqual[N[(y * z), $MachinePrecision], 1e+130], N[(0.125 * x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -2.00000000000000002e55 or 1.0000000000000001e130 < (*.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 y around inf

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

      1. *-commutative N/A

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

      2. associate-*l* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 88.5

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

   5. Applied rewrites 88.5%

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

   if -2.00000000000000002e55 < (*.f64 y z) < 1.0000000000000001e130

   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 y 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. lower-fma.f64 86.8

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

   5. Applied rewrites 86.8%

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

Alternative 4: 64.2% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(0.125 * x + 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 y 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. lower-fma.f64 57.1

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

5. Applied rewrites 57.1%

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

Alternative 5: 32.5% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

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 * 0.125d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(x * 0.125), $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. lower-*.f64 27.8

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

5. Applied rewrites 27.8%

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

6. Final simplification 27.8%

   \[\leadsto x \cdot 0.125 \]
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 2024232 
(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))