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

Percentage Accurate: 100.0% → 100.0%

Time: 3.9s

Alternatives: 6

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: 84.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{1}{8}\\ t_2 := \mathsf{fma}\left(-0.5 \cdot y, z, 0.125 \cdot x\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+86}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ 1.0 8.0))) (t_2 (fma (* -0.5 y) z (* 0.125 x))))
   (if (<= t_1 -2e+86) t_2 (if (<= t_1 0.05) (fma (* -0.5 y) z t) t_2))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (1.0 / 8.0);
	double t_2 = fma((-0.5 * y), z, (0.125 * x));
	double tmp;
	if (t_1 <= -2e+86) {
		tmp = t_2;
	} else if (t_1 <= 0.05) {
		tmp = fma((-0.5 * y), z, t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(1.0 / 8.0))
	t_2 = fma(Float64(-0.5 * y), z, Float64(0.125 * x))
	tmp = 0.0
	if (t_1 <= -2e+86)
		tmp = t_2;
	elseif (t_1 <= 0.05)
		tmp = fma(Float64(-0.5 * y), z, t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(1.0 / 8.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(-0.5 * y), $MachinePrecision] * z + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+86], t$95$2, If[LessEqual[t$95$1, 0.05], N[(N[(-0.5 * y), $MachinePrecision] * z + t), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 8 binary64)) x) < -2e86 or 0.050000000000000003 < (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 8 binary64)) x)

   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 t around 0

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

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

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

      2. metadata-eval N/A

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

      3. +-commutative N/A

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

      4. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -2e86 < (*.f64 (/.f64 #s(literal 1 binary64) #s(literal 8 binary64)) x) < 0.050000000000000003

   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. associate-*r* N/A

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

      5. lower-fma.f64 N/A

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

      6. lower-*.f64 87.2

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

   5. Applied rewrites 87.2%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5 \cdot y, z, t\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot \frac{1}{8} \leq -2 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y, z, 0.125 \cdot x\right)\\ \mathbf{elif}\;x \cdot \frac{1}{8} \leq 0.05:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y, z, 0.125 \cdot x\right)\\ \end{array} \]
5. 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 2024230 
(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))