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

Percentage Accurate: 99.9% → 99.9%

Time: 5.6s

Alternatives: 7

Speedup: N/A×

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: 99.9% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.7%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
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. distribute-neg-frac2 N/A

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

   8. div-inv N/A

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

   9. +-commutative N/A

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

   10. lower-fma.f64 N/A

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

   11. lift-*.f64 N/A

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

   12. *-commutative N/A

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

   13. lower-*.f64 N/A

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

   14. metadata-eval N/A

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

   15. metadata-eval N/A

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

   16. +-commutative N/A

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

   17. lift-*.f64 N/A

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

   18. *-commutative N/A

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

   19. lower-fma.f64 99.7

       \[\leadsto \mathsf{fma}\left(z \cdot y, -0.5, \color{blue}{\mathsf{fma}\left(x, \frac{1}{8}, t\right)}\right) \]

   20. lift-/.f64 N/A

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

   21. metadata-eval 99.7

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

4. Applied rewrites 99.7%

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

Alternative 2: 87.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -4 \cdot 10^{+28} \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+31}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, 0.125 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y z) -4e+28) (not (<= (* y z) 2e+31)))
   (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 (((y * z) <= -4e+28) || !((y * z) <= 2e+31)) {
		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 ((Float64(y * z) <= -4e+28) || !(Float64(y * z) <= 2e+31))
		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[Or[LessEqual[N[(y * z), $MachinePrecision], -4e+28], N[Not[LessEqual[N[(y * z), $MachinePrecision], 2e+31]], $MachinePrecision]], N[(N[(-0.5 * z), $MachinePrecision] * y + N[(0.125 * x), $MachinePrecision]), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -3.99999999999999983e28 or 1.9999999999999999e31 < (*.f64 y z)

   1. Initial program 99.4%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   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 32.6

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

   5. Applied rewrites 32.6%

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

   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

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 N/A

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

      12. lower-*.f64 88.2

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

   8. Applied rewrites 88.2%

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

   if -3.99999999999999983e28 < (*.f64 y z) < 1.9999999999999999e31

   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 95.1

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

   5. Applied rewrites 95.1%

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

4. Final simplification 91.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -4 \cdot 10^{+28} \lor \neg \left(y \cdot z \leq 2 \cdot 10^{+31}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot z, y, 0.125 \cdot x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+76} \lor \neg \left(y \cdot z \leq 5 \cdot 10^{+48}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y z) -1e+76) (not (<= (* y z) 5e+48)))
   (fma (* -0.5 y) z t)
   (fma 0.125 x t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * z) <= -1e+76) || !((y * z) <= 5e+48)) {
		tmp = fma((-0.5 * y), z, t);
	} else {
		tmp = fma(0.125, x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(y * z) <= -1e+76) || !(Float64(y * z) <= 5e+48))
		tmp = fma(Float64(-0.5 * y), z, t);
	else
		tmp = fma(0.125, x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], -1e+76], N[Not[LessEqual[N[(y * z), $MachinePrecision], 5e+48]], $MachinePrecision]], N[(N[(-0.5 * y), $MachinePrecision] * z + t), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -1e76 or 4.99999999999999973e48 < (*.f64 y z)

   1. Initial program 99.3%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   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 82.0

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

   5. Applied rewrites 82.0%

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

   if -1e76 < (*.f64 y z) < 4.99999999999999973e48

   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 91.6

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

   5. Applied rewrites 91.6%

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

4. Final simplification 87.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1 \cdot 10^{+76} \lor \neg \left(y \cdot z \leq 5 \cdot 10^{+48}\right):\\ \;\;\;\;\mathsf{fma}\left(-0.5 \cdot y, z, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+156} \lor \neg \left(y \cdot z \leq 5 \cdot 10^{+240}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* y z) -2e+156) (not (<= (* y z) 5e+240)))
   (* -0.5 (* z y))
   (fma 0.125 x t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((y * z) <= -2e+156) || !((y * z) <= 5e+240)) {
		tmp = -0.5 * (z * y);
	} else {
		tmp = fma(0.125, x, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(y * z) <= -2e+156) || !(Float64(y * z) <= 5e+240))
		tmp = Float64(-0.5 * Float64(z * y));
	else
		tmp = fma(0.125, x, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(y * z), $MachinePrecision], -2e+156], N[Not[LessEqual[N[(y * z), $MachinePrecision], 5e+240]], $MachinePrecision]], N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision], N[(0.125 * x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -2e156 or 5.0000000000000003e240 < (*.f64 y z)

   1. Initial program 98.8%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   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 11.0

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

   5. Applied rewrites 11.0%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(y \cdot z\right)} \]
   7. 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 86.8

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

   8. Applied rewrites 86.8%

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

   if -2e156 < (*.f64 y z) < 5.0000000000000003e240

   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 81.5

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

   5. Applied rewrites 81.5%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+156} \lor \neg \left(y \cdot z \leq 5 \cdot 10^{+240}\right):\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \cdot z \leq -2 \cdot 10^{+156}:\\ \;\;\;\;-0.5 \cdot \left(z \cdot y\right)\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+240}:\\ \;\;\;\;\mathsf{fma}\left(0.125, x, t\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot -0.5\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (* y z) -2e+156)
   (* -0.5 (* z y))
   (if (<= (* y z) 5e+240) (fma 0.125 x t) (* (* y -0.5) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y * z) <= -2e+156) {
		tmp = -0.5 * (z * y);
	} else if ((y * z) <= 5e+240) {
		tmp = fma(0.125, x, t);
	} else {
		tmp = (y * -0.5) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(y * z) <= -2e+156)
		tmp = Float64(-0.5 * Float64(z * y));
	elseif (Float64(y * z) <= 5e+240)
		tmp = fma(0.125, x, t);
	else
		tmp = Float64(Float64(y * -0.5) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(y * z), $MachinePrecision], -2e+156], N[(-0.5 * N[(z * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(y * z), $MachinePrecision], 5e+240], N[(0.125 * x + t), $MachinePrecision], N[(N[(y * -0.5), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -2e156

   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 13.9

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

   5. Applied rewrites 13.9%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(y \cdot z\right)} \]
   7. 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 85.8

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

   8. Applied rewrites 85.8%

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

   if -2e156 < (*.f64 y z) < 5.0000000000000003e240

   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 81.5

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

   5. Applied rewrites 81.5%

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

   if 5.0000000000000003e240 < (*.f64 y z)

   1. Initial program 96.9%

      \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
   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 6.6

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

   5. Applied rewrites 6.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{-1}{2} \cdot \left(y \cdot z\right)} \]
   7. 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 88.3

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

   8. Applied rewrites 88.3%

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

   10. Applied rewrites 91.4%

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

Alternative 6: 63.9% 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 99.7%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
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 63.6

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

5. Applied rewrites 63.6%

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

Alternative 7: 32.7% 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 99.7%

   \[\left(\frac{1}{8} \cdot x - \frac{y \cdot z}{2}\right) + t \]
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 63.6

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

5. Applied rewrites 63.6%

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

6. Taylor expanded in x around inf

   \[\leadsto \color{blue}{\frac{1}{8} \cdot x} \]
7. Step-by-step derivation

   1. lower-*.f64 32.5

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

8. Applied rewrites 32.5%

   \[\leadsto \color{blue}{0.125 \cdot x} \]
9. 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 2024313 
(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))