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

Percentage Accurate: 100.0% → 100.0%

Time: 4.3s

Alternatives: 4

Speedup: 0.1×

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

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

   1. associate-+l- 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-+l- 99.7%

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

   4. metadata-eval 99.7%

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

   5. *-commutative 99.7%

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

   6. 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. Taylor expanded in x around 0 99.7%

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

   1. associate-*r* 100.0%

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

   2. fma-def 100.0%

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

   3. metadata-eval 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.5}{-1}} \cdot y, z, 0.125 \cdot x\right) + t \]

   4. associate-/r/ 99.9%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.5}{\frac{-1}{y}}}, z, 0.125 \cdot x\right) + t \]

   5. associate-/l* 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{0.5 \cdot y}{-1}}, z, 0.125 \cdot x\right) + t \]

   6. *-commutative 100.0%

      \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{y \cdot 0.5}}{-1}, z, 0.125 \cdot x\right) + t \]

   7. associate-/l* 100.0%

      \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{\frac{-1}{0.5}}}, z, 0.125 \cdot x\right) + t \]

   8. metadata-eval 100.0%

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

   9. fma-def 100.0%

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

   10. *-commutative 100.0%

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

   11. fma-def 100.0%

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

6. Simplified 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 79.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-108} \lor \neg \left(z \leq 2.15 \cdot 10^{+71}\right):\\ \;\;\;\;t + z \cdot \frac{y}{-2}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.5e-108) (not (<= z 2.15e+71)))
   (+ t (* z (/ y -2.0)))
   (+ (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e-108) || !(z <= 2.15e+71)) {
		tmp = t + (z * (y / -2.0));
	} else {
		tmp = (0.125 * x) + t;
	}
	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 ((z <= (-5.5d-108)) .or. (.not. (z <= 2.15d+71))) then
        tmp = t + (z * (y / (-2.0d0)))
    else
        tmp = (0.125d0 * x) + t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.5e-108) || !(z <= 2.15e+71)) {
		tmp = t + (z * (y / -2.0));
	} else {
		tmp = (0.125 * x) + t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.5e-108) or not (z <= 2.15e+71):
		tmp = t + (z * (y / -2.0))
	else:
		tmp = (0.125 * x) + t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.5e-108) || !(z <= 2.15e+71))
		tmp = Float64(t + Float64(z * Float64(y / -2.0)));
	else
		tmp = Float64(Float64(0.125 * x) + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.5e-108) || ~((z <= 2.15e+71)))
		tmp = t + (z * (y / -2.0));
	else
		tmp = (0.125 * x) + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.5e-108], N[Not[LessEqual[z, 2.15e+71]], $MachinePrecision]], N[(t + N[(z * N[(y / -2.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * x), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.50000000000000031e-108 or 2.14999999999999992e71 < z

   1. Initial program 99.4%

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

      1. associate-+l- 99.4%

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

      2. *-commutative 99.4%

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

      3. associate-+l- 99.4%

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

      4. metadata-eval 99.4%

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

      5. *-commutative 99.4%

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

      6. 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. Taylor expanded in x around 0 81.6%

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

      1. associate-*r* 82.1%

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

      2. *-commutative 82.1%

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

      3. metadata-eval 82.1%

         \[\leadsto z \cdot \left(\color{blue}{\frac{0.5}{-1}} \cdot y\right) + t \]

      4. associate-/r/ 82.0%

         \[\leadsto z \cdot \color{blue}{\frac{0.5}{\frac{-1}{y}}} + t \]

      5. associate-/l* 82.1%

         \[\leadsto z \cdot \color{blue}{\frac{0.5 \cdot y}{-1}} + t \]

      6. *-commutative 82.1%

         \[\leadsto z \cdot \frac{\color{blue}{y \cdot 0.5}}{-1} + t \]

      7. associate-/l* 82.1%

         \[\leadsto z \cdot \color{blue}{\frac{y}{\frac{-1}{0.5}}} + t \]

      8. metadata-eval 82.1%

         \[\leadsto z \cdot \frac{y}{\color{blue}{-2}} + t \]

   6. Simplified 82.1%

      \[\leadsto \color{blue}{z \cdot \frac{y}{-2}} + t \]

   if -5.50000000000000031e-108 < z < 2.14999999999999992e71

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. 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. Taylor expanded in x around inf 80.9%

      \[\leadsto \color{blue}{0.125 \cdot x} + t \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{-108} \lor \neg \left(z \leq 2.15 \cdot 10^{+71}\right):\\ \;\;\;\;t + z \cdot \frac{y}{-2}\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + t\\ \end{array} \]

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 99.7%

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

   1. associate-+l- 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-+l- 99.7%

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

   4. metadata-eval 99.7%

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

   5. *-commutative 99.7%

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

   6. 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. Final simplification 99.9%

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

Alternative 4: 64.3% 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 99.7%

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

   1. associate-+l- 99.7%

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

   2. *-commutative 99.7%

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

   3. associate-+l- 99.7%

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

   4. metadata-eval 99.7%

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

   5. *-commutative 99.7%

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

   6. 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. Taylor expanded in x around inf 58.8%

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

5. Final simplification 58.8%

   \[\leadsto 0.125 \cdot x + t \]

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 2023301 
(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))