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

Percentage Accurate: 100.0% → 100.0%

Time: 4.5s

Alternatives: 4

Speedup: 2.2×

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, 2.0× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, z \cdot y, 0.125 \cdot x\right) + t \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ (fma -0.5 (* z y) (* 0.125 x)) t))

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(-0.5 * N[(z * y), $MachinePrecision] + N[(0.125 * x), $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. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x - \frac{1}{2} \cdot \left(y \cdot z\right)\right)} + t \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{\left(\frac{1}{8} \cdot x + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(y \cdot z\right)\right)} + t \]

   2. metadata-eval N/A

      \[\leadsto \left(\frac{1}{8} \cdot x + \color{blue}{\frac{-1}{2}} \cdot \left(y \cdot z\right)\right) + t \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \left(y \cdot z\right) + \frac{1}{8} \cdot x\right)} + t \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, y \cdot z, \frac{1}{8} \cdot x\right)} + t \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{z \cdot y}, \frac{1}{8} \cdot x\right) + t \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{z \cdot y}, \frac{1}{8} \cdot x\right) + t \]

   7. lower-*.f64 100.0

      \[\leadsto \mathsf{fma}\left(-0.5, z \cdot y, \color{blue}{0.125 \cdot x}\right) + t \]

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, z \cdot y, 0.125 \cdot x\right)} + t \]
6. Add Preprocessing

Alternative 2: 86.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+162} \lor \neg \left(y \cdot z \leq 4 \cdot 10^{-21}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y z) -1e+162) (not (<= (* y z) 4e-21)))
   (fma -0.5 (* z y) t)
   (fma 0.125 x t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * z) <= -1e+162) || !((y * z) <= 4e-21)) {
		tmp = fma(-0.5, (z * y), t);
	} else {
		tmp = fma(0.125, x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(y * z) <= -1e+162) || !(Float64(y * z) <= 4e-21))
		tmp = fma(-0.5, Float64(z * y), t);
	else
		tmp = fma(0.125, x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], -1e+162], N[Not[LessEqual[N[(y * z), $MachinePrecision], 4e-21]], $MachinePrecision]], N[(-0.5 * N[(z * y), $MachinePrecision] + t), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -9.9999999999999994e161 or 3.99999999999999963e-21 < (*.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. fp-cancel-sub-sign-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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 88.3

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

   5. Applied rewrites 88.3%

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

   if -9.9999999999999994e161 < (*.f64 y z) < 3.99999999999999963e-21

   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 89.8

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

   5. Applied rewrites 89.8%

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

4. Final simplification 89.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+162} \lor \neg \left(y \cdot z \leq 4 \cdot 10^{-21}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5, z \cdot y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 100.0% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(-0.5, z \cdot y, \mathsf{fma}\left(0.125, x, t\right)\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(-0.5 * N[(z * y), $MachinePrecision] + N[(0.125 * x + t), $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. Taylor expanded in x around 0

   \[\leadsto \color{blue}{\left(t + \frac{1}{8} \cdot x\right) - \frac{1}{2} \cdot \left(y \cdot z\right)} \]
4. Step-by-step derivation

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

      \[\leadsto \color{blue}{\left(t + \frac{1}{8} \cdot x\right) + \left(\mathsf{neg}\left(\frac{1}{2}\right)\right) \cdot \left(y \cdot z\right)} \]

   2. metadata-eval N/A

      \[\leadsto \left(t + \frac{1}{8} \cdot x\right) + \color{blue}{\frac{-1}{2}} \cdot \left(y \cdot z\right) \]

   3. +-commutative N/A

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(y \cdot z\right) + \left(t + \frac{1}{8} \cdot x\right)} \]

   4. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2}, y \cdot z, t + \frac{1}{8} \cdot x\right)} \]

   5. *-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{z \cdot y}, t + \frac{1}{8} \cdot x\right) \]

   6. lower-*.f64 N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, \color{blue}{z \cdot y}, t + \frac{1}{8} \cdot x\right) \]

   7. +-commutative N/A

      \[\leadsto \mathsf{fma}\left(\frac{-1}{2}, z \cdot y, \color{blue}{\frac{1}{8} \cdot x + t}\right) \]

   8. lower-fma.f64 100.0

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

5. Applied rewrites 100.0%

   \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, z \cdot y, \mathsf{fma}\left(0.125, x, t\right)\right)} \]
6. 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 68.5

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

5. Applied rewrites 68.5%

   \[\leadsto \color{blue}{\mathsf{fma}\left(0.125, x, t\right)} \]
6. 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 2024329 
(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))