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

Percentage Accurate: 100.0% → 100.0%

Time: 3.1s

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: 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, 2.2× 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: 88.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y \cdot -0.5, z, t\right)\\ \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot z \leq 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (* y -0.5) z t)))
   (if (<= (* y z) -1e+100) t_1 (if (<= (* y z) 1e+27) (fma 0.125 x t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((y * -0.5), z, t);
	double tmp;
	if ((y * z) <= -1e+100) {
		tmp = t_1;
	} else if ((y * z) <= 1e+27) {
		tmp = fma(0.125, x, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(y * -0.5), z, t)
	tmp = 0.0
	if (Float64(y * z) <= -1e+100)
		tmp = t_1;
	elseif (Float64(y * z) <= 1e+27)
		tmp = fma(0.125, x, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * -0.5), $MachinePrecision] * z + t), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -1e+100], t$95$1, If[LessEqual[N[(y * z), $MachinePrecision], 1e+27], N[(0.125 * x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.00000000000000002e100 or 1e27 < (*.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. 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 88.2

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

   5. Applied rewrites 88.2%

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

   if -1.00000000000000002e100 < (*.f64 y z) < 1e27

   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 90.0

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

   5. Applied rewrites 90.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+100}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.5, z, t\right)\\ \mathbf{elif}\;y \cdot z \leq 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.5, z, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 84.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot z\right) \cdot -0.5\\ \mathbf{if}\;y \cdot z \leq -1.6 \cdot 10^{+97}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot z \leq 1.5 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* y z) -0.5)))
   (if (<= (* y z) -1.6e+97)
     t_1
     (if (<= (* y z) 1.5e+98) (fma 0.125 x 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) <= -1.6e+97) {
		tmp = t_1;
	} else if ((y * z) <= 1.5e+98) {
		tmp = fma(0.125, x, 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) <= -1.6e+97)
		tmp = t_1;
	elseif (Float64(y * z) <= 1.5e+98)
		tmp = fma(0.125, x, 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], -1.6e+97], t$95$1, If[LessEqual[N[(y * z), $MachinePrecision], 1.5e+98], N[(0.125 * x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1.60000000000000008e97 or 1.5000000000000001e98 < (*.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. 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 y around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 83.8

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

   7. Applied rewrites 83.8%

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

   if -1.60000000000000008e97 < (*.f64 y z) < 1.5000000000000001e98

   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 87.4

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

   5. Applied rewrites 87.4%

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

4. Final simplification 86.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1.6 \cdot 10^{+97}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -0.5\\ \mathbf{elif}\;y \cdot z \leq 1.5 \cdot 10^{+98}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.5, z, t\right)\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, 0.125 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -8e+90)
   (fma (* y -0.5) z t)
   (if (<= t 9.8e-8) (fma (* -0.5 z) y (* 0.125 x)) (fma 0.125 x t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8e+90) {
		tmp = fma((y * -0.5), z, t);
	} else if (t <= 9.8e-8) {
		tmp = fma((-0.5 * z), y, (0.125 * x));
	} else {
		tmp = fma(0.125, x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8e+90)
		tmp = fma(Float64(y * -0.5), z, t);
	elseif (t <= 9.8e-8)
		tmp = fma(Float64(-0.5 * z), y, Float64(0.125 * x));
	else
		tmp = fma(0.125, x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -8e+90], N[(N[(y * -0.5), $MachinePrecision] * z + t), $MachinePrecision], If[LessEqual[t, 9.8e-8], N[(N[(-0.5 * z), $MachinePrecision] * y + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.99999999999999973e90

   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 95.4

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

   5. Applied rewrites 95.4%

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

   if -7.99999999999999973e90 < t < 9.8000000000000004e-8

   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. lower-*.f64 45.8

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

   5. Applied rewrites 45.8%

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

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{1}{8} \cdot x - \frac{1}{2} \cdot \left(y \cdot z\right)} \]
   7. 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. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-*.f64 92.3

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

   8. Applied rewrites 92.3%

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

   if 9.8000000000000004e-8 < t

   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 83.7

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

   5. Applied rewrites 83.7%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+90}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot -0.5, z, t\right)\\ \mathbf{elif}\;t \leq 9.8 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, 0.125 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 100.0% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(x * 0.125 + N[(N[(-0.5 * z), $MachinePrecision] * y + 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. associate-+l+ N/A

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

   5. lift-*.f64 N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

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

   8. lift-/.f64 N/A

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

   9. metadata-eval N/A

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

   10. lift-/.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. associate-/l* N/A

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

   13. *-commutative N/A

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

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

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

   15. lower-fma.f64 N/A

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

   16. div-inv N/A

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

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

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

   18. metadata-eval N/A

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

   19. metadata-eval N/A

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

   20. metadata-eval N/A

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

   21. metadata-eval N/A

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

   22. lower-*.f64 N/A

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

   23. metadata-eval N/A

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

   24. metadata-eval 100.0

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

4. Applied rewrites 100.0%

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

5. Final simplification 100.0%

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

Alternative 6: 64.8% 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 64.7

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

5. Applied rewrites 64.7%

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

Alternative 7: 33.0% 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. lower-*.f64 31.3

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

5. Applied rewrites 31.3%

   \[\leadsto \color{blue}{0.125 \cdot x} \]
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 2024304 
(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))