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

Percentage Accurate: 100.0% → 100.0%

Time: 6.0s

Alternatives: 5

Speedup: 1.2×

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(y * N[(z * -0.5), $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. Step-by-step derivation

   1. metadata-eval 100.0%

      \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. associate-/l* 99.8%

      \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\left(-0.5 \cdot \left(y \cdot z\right) + 0.125 \cdot x\right)} + t \]
6. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. metadata-eval 100.0%

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

   4. distribute-rgt-neg-in 100.0%

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

   5. fma-udef 100.0%

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

   6. distribute-rgt-neg-in 100.0%

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

   7. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 2: 82.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+88} \lor \neg \left(x \leq 2 \cdot 10^{+18}\right):\\ \;\;\;\;0.125 \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot \left(z \cdot -0.5\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -8.5e+88) (not (<= x 2e+18)))
   (+ (* 0.125 x) t)
   (+ t (* y (* z -0.5)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.5e+88) || !(x <= 2e+18)) {
		tmp = (0.125 * x) + t;
	} else {
		tmp = t + (y * (z * -0.5));
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((x <= (-8.5d+88)) .or. (.not. (x <= 2d+18))) then
        tmp = (0.125d0 * x) + t
    else
        tmp = t + (y * (z * (-0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -8.5e+88) || !(x <= 2e+18)) {
		tmp = (0.125 * x) + t;
	} else {
		tmp = t + (y * (z * -0.5));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -8.5e+88) or not (x <= 2e+18):
		tmp = (0.125 * x) + t
	else:
		tmp = t + (y * (z * -0.5))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -8.5e+88) || !(x <= 2e+18))
		tmp = Float64(Float64(0.125 * x) + t);
	else
		tmp = Float64(t + Float64(y * Float64(z * -0.5)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -8.5e+88) || ~((x <= 2e+18)))
		tmp = (0.125 * x) + t;
	else
		tmp = t + (y * (z * -0.5));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -8.5e+88], N[Not[LessEqual[x, 2e+18]], $MachinePrecision]], N[(N[(0.125 * x), $MachinePrecision] + t), $MachinePrecision], N[(t + N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -8.5000000000000005e88 or 2e18 < x

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.9%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 81.9%

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

   if -8.5000000000000005e88 < x < 2e18

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.8%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 89.1%

      \[\leadsto \color{blue}{-0.5 \cdot \left(y \cdot z\right)} + t \]
   6. Step-by-step derivation

      1. *-commutative 89.1%

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

      2. associate-*l* 89.1%

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

   7. Simplified 89.1%

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

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5 \cdot 10^{+88} \lor \neg \left(x \leq 2 \cdot 10^{+18}\right):\\ \;\;\;\;0.125 \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;t + y \cdot \left(z \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ t + \left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ t (- (* 0.125 x) (/ y (/ 2.0 z)))))

C

double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - (y / (2.0 / z)));
}

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 + ((0.125d0 * x) - (y / (2.0d0 / z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - (y / (2.0 / z)));
}

Python

def code(x, y, z, t):
	return t + ((0.125 * x) - (y / (2.0 / z)))

Julia

function code(x, y, z, t)
	return Float64(t + Float64(Float64(0.125 * x) - Float64(y / Float64(2.0 / z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t + ((0.125 * x) - (y / (2.0 / z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(t + N[(N[(0.125 * x), $MachinePrecision] - N[(y / N[(2.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. associate-/l* 99.8%

      \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
4. Add Preprocessing

5. Final simplification 99.8%

   \[\leadsto t + \left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) \]
6. Add Preprocessing

Alternative 4: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ t + \left(0.125 \cdot x - \frac{y \cdot z}{2}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ t (- (* 0.125 x) (/ (* y z) 2.0))))

C

double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - ((y * z) / 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 + ((0.125d0 * x) - ((y * z) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - ((y * z) / 2.0));
}

Python

def code(x, y, z, t):
	return t + ((0.125 * x) - ((y * z) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(t + Float64(Float64(0.125 * x) - Float64(Float64(y * z) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t + ((0.125 * x) - ((y * z) / 2.0));
end

Wolfram

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

   \[\leadsto t + \left(0.125 \cdot x - \frac{y \cdot z}{2}\right) \]
4. Add Preprocessing

Alternative 5: 64.2% accurate, 2.6× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (0.125 * x) + 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 = (0.125d0 * x) + t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. associate-/l* 99.8%

      \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

3. Simplified 99.8%

   \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 62.1%

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

6. Final simplification 62.1%

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

Developer target: 100.0% accurate, 1.2× 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 2024040 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (+ (/ x 8.0) t) (* (/ z 2.0) y))

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