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

Percentage Accurate: 99.9% → 100.0%

Time: 4.5s

Alternatives: 7

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: 99.9% 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, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(-0.5 * z), $MachinePrecision] * y + N[(x * 0.125 + 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. Step-by-step derivation

   1. lift-+.f64 N/A

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

   2. lift--.f64 N/A

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

   3. sub-neg N/A

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

   4. +-commutative N/A

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

   5. associate-+l+ N/A

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

   6. lift-/.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. associate-/l* N/A

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

   9. *-commutative N/A

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

   10. distribute-lft-neg-in N/A

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

   11. +-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. div-inv N/A

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

   14. distribute-rgt-neg-in N/A

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

   15. metadata-eval N/A

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

   16. metadata-eval N/A

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

   17. metadata-eval N/A

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

   18. metadata-eval N/A

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

   19. lower-*.f64 N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. +-commutative N/A

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

   23. lift-*.f64 N/A

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

   24. *-commutative N/A

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, 0.125 \cdot x\right)\\ \mathbf{elif}\;y \cdot z \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y z) -2e-65)
   (fma (* -0.5 z) y (* 0.125 x))
   (if (<= (* y z) 0.0001) (fma x 0.125 t) (fma -0.5 (* y z) (* 0.125 x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * z) <= -2e-65) {
		tmp = fma((-0.5 * z), y, (0.125 * x));
	} else if ((y * z) <= 0.0001) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = fma(-0.5, (y * z), (0.125 * x));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * z) <= -2e-65)
		tmp = fma(Float64(-0.5 * z), y, Float64(0.125 * x));
	elseif (Float64(y * z) <= 0.0001)
		tmp = fma(x, 0.125, t);
	else
		tmp = fma(-0.5, Float64(y * z), Float64(0.125 * x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * z), $MachinePrecision], -2e-65], N[(N[(-0.5 * z), $MachinePrecision] * y + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 0.0001], N[(x * 0.125 + t), $MachinePrecision], N[(-0.5 * N[(y * z), $MachinePrecision] + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -1.99999999999999985e-65

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. associate-+l+ N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

      9. *-commutative N/A

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

      10. distribute-lft-neg-in N/A

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

      11. +-commutative N/A

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

      12. lower-fma.f64 N/A

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

      13. div-inv N/A

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

      14. distribute-rgt-neg-in N/A

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

      15. metadata-eval N/A

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

      16. metadata-eval N/A

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

      17. metadata-eval N/A

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

      18. metadata-eval N/A

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

      19. lower-*.f64 N/A

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

      20. metadata-eval N/A

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

      21. metadata-eval N/A

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

      22. +-commutative N/A

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

      23. lift-*.f64 N/A

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

      24. *-commutative N/A

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in t around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 84.0

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

   7. Applied rewrites 84.0%

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

   if -1.99999999999999985e-65 < (*.f64 y z) < 1.00000000000000005e-4

   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 z 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. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{8}} + t \]

      3. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

   if 1.00000000000000005e-4 < (*.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 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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 85.7

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

   5. Applied rewrites 85.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(-0.5, z \cdot y, x \cdot 0.125\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, 0.125 \cdot x\right)\\ \mathbf{elif}\;y \cdot z \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)\\ \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{-65}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot z \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma -0.5 (* y z) (* 0.125 x))))
   (if (<= (* y z) -2e-65) t_1 (if (<= (* y z) 0.0001) (fma x 0.125 t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(-0.5, (y * z), (0.125 * x));
	double tmp;
	if ((y * z) <= -2e-65) {
		tmp = t_1;
	} else if ((y * z) <= 0.0001) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(-0.5, Float64(y * z), Float64(0.125 * x))
	tmp = 0.0
	if (Float64(y * z) <= -2e-65)
		tmp = t_1;
	elseif (Float64(y * z) <= 0.0001)
		tmp = fma(x, 0.125, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(-0.5 * N[(y * z), $MachinePrecision] + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -2e-65], t$95$1, If[LessEqual[N[(y * z), $MachinePrecision], 0.0001], N[(x * 0.125 + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.99999999999999985e-65 or 1.00000000000000005e-4 < (*.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 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. lower-fma.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 84.7

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

   5. Applied rewrites 84.7%

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

   if -1.99999999999999985e-65 < (*.f64 y z) < 1.00000000000000005e-4

   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 z 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. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{8}} + t \]

      3. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{-65}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)\\ \mathbf{elif}\;y \cdot z \leq 0.0001:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-0.5, y \cdot z, 0.125 \cdot x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-0.5 \cdot z, y, t\right)\\ \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (* -0.5 z) y t)))
   (if (<= (* y z) -1e+40) t_1 (if (<= (* y z) 5e+33) (fma x 0.125 t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((-0.5 * z), y, t);
	double tmp;
	if ((y * z) <= -1e+40) {
		tmp = t_1;
	} else if ((y * z) <= 5e+33) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(-0.5 * z), y, t)
	tmp = 0.0
	if (Float64(y * z) <= -1e+40)
		tmp = t_1;
	elseif (Float64(y * z) <= 5e+33)
		tmp = fma(x, 0.125, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(-0.5 * z), $MachinePrecision] * y + t), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -1e+40], t$95$1, If[LessEqual[N[(y * z), $MachinePrecision], 5e+33], N[(x * 0.125 + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.00000000000000003e40 or 4.99999999999999973e33 < (*.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

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 83.0

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

   5. Applied rewrites 83.0%

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

   if -1.00000000000000003e40 < (*.f64 y z) < 4.99999999999999973e33

   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 z 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. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{8}} + t \]

      3. lower-fma.f64 93.3

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

   5. Applied rewrites 93.3%

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

Alternative 5: 84.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot z\right) \cdot -0.5\\ \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+141}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot z \leq 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* y z) -0.5)))
   (if (<= (* y z) -1e+141) t_1 (if (<= (* y z) 1e+144) (fma x 0.125 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) <= -1e+141) {
		tmp = t_1;
	} else if ((y * z) <= 1e+144) {
		tmp = fma(x, 0.125, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) * -0.5)
	tmp = 0.0
	if (Float64(y * z) <= -1e+141)
		tmp = t_1;
	elseif (Float64(y * z) <= 1e+144)
		tmp = fma(x, 0.125, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -1e+141], t$95$1, If[LessEqual[N[(y * z), $MachinePrecision], 1e+144], N[(x * 0.125 + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.00000000000000002e141 or 1.00000000000000002e144 < (*.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 z around inf

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

      1. lower-*.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 81.3

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

   5. Applied rewrites 81.3%

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

   if -1.00000000000000002e141 < (*.f64 y z) < 1.00000000000000002e144

   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 z 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. *-commutative N/A

         \[\leadsto \color{blue}{x \cdot \frac{1}{8}} + t \]

      3. lower-fma.f64 86.1

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

   5. Applied rewrites 86.1%

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

4. Final simplification 84.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+141}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -0.5\\ \mathbf{elif}\;y \cdot z \leq 10^{+144}:\\ \;\;\;\;\mathsf{fma}\left(x, 0.125, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.4% accurate, 5.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(x * 0.125 + 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 z 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. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{8}} + t \]

   3. lower-fma.f64 66.2

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

5. Applied rewrites 66.2%

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

Alternative 7: 32.9% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(0.125 * x), $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. *-commutative N/A

      \[\leadsto \color{blue}{x \cdot \frac{1}{8}} \]

   2. lower-*.f64 35.7

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

5. Applied rewrites 35.7%

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

6. Final simplification 35.7%

   \[\leadsto 0.125 \cdot x \]
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 2024268 
(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))