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

Percentage Accurate: 97.8% → 98.3%

Time: 9.5s

Alternatives: 11

Speedup: 1.4×

• 33.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Initial Program: 97.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Fortran

real(8) function code(x, y, z, t, a, b, c)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8), intent (in) :: b
    real(8), intent (in) :: c
    code = (((x * y) + ((z * t) / 16.0d0)) - ((a * b) / 4.0d0)) + c
end function

Java

public static double code(double x, double y, double z, double t, double a, double b, double c) {
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
}

Python

def code(x, y, z, t, a, b, c):
	return (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c

Julia

function code(x, y, z, t, a, b, c)
	return Float64(Float64(Float64(Float64(x * y) + Float64(Float64(z * t) / 16.0)) - Float64(Float64(a * b) / 4.0)) + c)
end

MATLAB

function tmp = code(x, y, z, t, a, b, c)
	tmp = (((x * y) + ((z * t) / 16.0)) - ((a * b) / 4.0)) + c;
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(N[(N[(x * y), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision] - N[(N[(a * b), $MachinePrecision] / 4.0), $MachinePrecision]), $MachinePrecision] + c), $MachinePrecision]

Alternative 1: 98.3% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y - \mathsf{fma}\left(a, b \cdot 0.25, -c\right)\right) \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (fma (* t 0.0625) z (- (* x y) (fma a (* b 0.25) (- c)))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	return fma((t * 0.0625), z, ((x * y) - fma(a, (b * 0.25), -c)));
}

Julia

function code(x, y, z, t, a, b, c)
	return fma(Float64(t * 0.0625), z, Float64(Float64(x * y) - fma(a, Float64(b * 0.25), Float64(-c))))
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := N[(N[(t * 0.0625), $MachinePrecision] * z + N[(N[(x * y), $MachinePrecision] - N[(a * N[(b * 0.25), $MachinePrecision] + (-c)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 99.2%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
2. Add Preprocessing
3. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

   2. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

   3. associate-+l- N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

   4. lift-+.f64 N/A

      \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

   5. +-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

   6. associate--l+ N/A

      \[\leadsto \color{blue}{\frac{z \cdot t}{16} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)} \]

   7. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

   8. lift-*.f64 N/A

      \[\leadsto \frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

   9. associate-/l* N/A

      \[\leadsto \color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

   10. *-commutative N/A

       \[\leadsto \color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

   11. lower-fma.f64 N/A

       \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)} \]

   12. div-inv N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

   13. lower-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

   14. metadata-eval N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

   15. lower--.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y - \left(\frac{a \cdot b}{4} - c\right)}\right) \]

   16. sub-neg N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, x \cdot y - \color{blue}{\left(\frac{a \cdot b}{4} + \left(\mathsf{neg}\left(c\right)\right)\right)}\right) \]

   17. lift-/.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, x \cdot y - \left(\color{blue}{\frac{a \cdot b}{4}} + \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

   18. lift-*.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, x \cdot y - \left(\frac{\color{blue}{a \cdot b}}{4} + \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

   19. associate-/l* N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, x \cdot y - \left(\color{blue}{a \cdot \frac{b}{4}} + \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

   20. lower-fma.f64 N/A

       \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, x \cdot y - \color{blue}{\mathsf{fma}\left(a, \frac{b}{4}, \mathsf{neg}\left(c\right)\right)}\right) \]

4. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y - \mathsf{fma}\left(a, b \cdot 0.25, -c\right)\right)} \]
5. Add Preprocessing

Alternative 2: 77.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y\right)\\ t_2 := x \cdot y + \frac{t \cdot z}{16}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+178}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma (* t 0.0625) z (* x y))) (t_2 (+ (* x y) (/ (* t z) 16.0))))
   (if (<= t_2 -5e+178) t_1 (if (<= t_2 1e+147) (fma a (* b -0.25) c) t_1))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma((t * 0.0625), z, (x * y));
	double t_2 = (x * y) + ((t * z) / 16.0);
	double tmp;
	if (t_2 <= -5e+178) {
		tmp = t_1;
	} else if (t_2 <= 1e+147) {
		tmp = fma(a, (b * -0.25), c);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(Float64(t * 0.0625), z, Float64(x * y))
	t_2 = Float64(Float64(x * y) + Float64(Float64(t * z) / 16.0))
	tmp = 0.0
	if (t_2 <= -5e+178)
		tmp = t_1;
	elseif (t_2 <= 1e+147)
		tmp = fma(a, Float64(b * -0.25), c);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(N[(t * 0.0625), $MachinePrecision] * z + N[(x * y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x * y), $MachinePrecision] + N[(N[(t * z), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+178], t$95$1, If[LessEqual[t$95$2, 1e+147], N[(a * N[(b * -0.25), $MachinePrecision] + c), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -4.9999999999999999e178 or 9.9999999999999998e146 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 95.5%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c} \]

      2. lift--.f64 N/A

         \[\leadsto \color{blue}{\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right)} + c \]

      3. associate-+l- N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right) - \left(\frac{a \cdot b}{4} - c\right)} \]

      4. lift-+.f64 N/A

         \[\leadsto \color{blue}{\left(x \cdot y + \frac{z \cdot t}{16}\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      5. +-commutative N/A

         \[\leadsto \color{blue}{\left(\frac{z \cdot t}{16} + x \cdot y\right)} - \left(\frac{a \cdot b}{4} - c\right) \]

      6. associate--l+ N/A

         \[\leadsto \color{blue}{\frac{z \cdot t}{16} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)} \]

      7. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{z \cdot t}{16}} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

      8. lift-*.f64 N/A

         \[\leadsto \frac{\color{blue}{z \cdot t}}{16} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

      9. associate-/l* N/A

         \[\leadsto \color{blue}{z \cdot \frac{t}{16}} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \color{blue}{\frac{t}{16} \cdot z} + \left(x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

      11. lower-fma.f64 N/A

          \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{t}{16}, z, x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right)} \]

      12. div-inv N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

      13. lower-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{t \cdot \frac{1}{16}}, z, x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

      14. metadata-eval N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \color{blue}{\frac{1}{16}}, z, x \cdot y - \left(\frac{a \cdot b}{4} - c\right)\right) \]

      15. lower--.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y - \left(\frac{a \cdot b}{4} - c\right)}\right) \]

      16. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, x \cdot y - \color{blue}{\left(\frac{a \cdot b}{4} + \left(\mathsf{neg}\left(c\right)\right)\right)}\right) \]

      17. lift-/.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, x \cdot y - \left(\color{blue}{\frac{a \cdot b}{4}} + \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      18. lift-*.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, x \cdot y - \left(\frac{\color{blue}{a \cdot b}}{4} + \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      19. associate-/l* N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, x \cdot y - \left(\color{blue}{a \cdot \frac{b}{4}} + \left(\mathsf{neg}\left(c\right)\right)\right)\right) \]

      20. lower-fma.f64 N/A

          \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, x \cdot y - \color{blue}{\mathsf{fma}\left(a, \frac{b}{4}, \mathsf{neg}\left(c\right)\right)}\right) \]

   4. Applied rewrites 96.4%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y - \mathsf{fma}\left(a, b \cdot 0.25, -c\right)\right)} \]

   5. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma}\left(t \cdot \frac{1}{16}, z, \color{blue}{x \cdot y}\right) \]
   6. Step-by-step derivation

      1. lower-*.f64 82.4

         \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, \color{blue}{x \cdot y}\right) \]

   7. Applied rewrites 82.4%

      \[\leadsto \mathsf{fma}\left(t \cdot 0.0625, z, \color{blue}{x \cdot y}\right) \]

   if -4.9999999999999999e178 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 9.9999999999999998e146

   1. Initial program 99.9%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\left(c + x \cdot y\right) - \frac{1}{4} \cdot \left(a \cdot b\right)} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

         \[\leadsto \color{blue}{\left(c + x \cdot y\right) + \left(\mathsf{neg}\left(\frac{1}{4}\right)\right) \cdot \left(a \cdot b\right)} \]

      2. metadata-eval N/A

         \[\leadsto \left(c + x \cdot y\right) + \color{blue}{\frac{-1}{4}} \cdot \left(a \cdot b\right) \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right) + \left(c + x \cdot y\right)} \]

      4. *-commutative N/A

         \[\leadsto \color{blue}{\left(a \cdot b\right) \cdot \frac{-1}{4}} + \left(c + x \cdot y\right) \]

      5. associate-*l* N/A

         \[\leadsto \color{blue}{a \cdot \left(b \cdot \frac{-1}{4}\right)} + \left(c + x \cdot y\right) \]

      6. *-commutative N/A

         \[\leadsto a \cdot \color{blue}{\left(\frac{-1}{4} \cdot b\right)} + \left(c + x \cdot y\right) \]

      7. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(a, \frac{-1}{4} \cdot b, c + x \cdot y\right)} \]

      8. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      9. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot \frac{-1}{4}}, c + x \cdot y\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(a, b \cdot \frac{-1}{4}, \color{blue}{x \cdot y + c}\right) \]

      11. lower-fma.f64 86.7

          \[\leadsto \mathsf{fma}\left(a, b \cdot -0.25, \color{blue}{\mathsf{fma}\left(x, y, c\right)}\right) \]

   5. Applied rewrites 86.7%

      \[\leadsto \color{blue}{\mathsf{fma}\left(a, b \cdot -0.25, \mathsf{fma}\left(x, y, c\right)\right)} \]

   6. Taylor expanded in x around 0

      \[\leadsto c + \color{blue}{\frac{-1}{4} \cdot \left(a \cdot b\right)} \]
   7. Step-by-step derivation

   8. Applied rewrites 73.6%

      \[\leadsto \mathsf{fma}\left(a, \color{blue}{b \cdot -0.25}, c\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y + \frac{t \cdot z}{16} \leq -5 \cdot 10^{+178}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y\right)\\ \mathbf{elif}\;x \cdot y + \frac{t \cdot z}{16} \leq 10^{+147}:\\ \;\;\;\;\mathsf{fma}\left(a, b \cdot -0.25, c\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t \cdot 0.0625, z, x \cdot y\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024228 
(FPCore (x y z t a b c)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, C"
  :precision binary64
  (+ (- (+ (* x y) (/ (* z t) 16.0)) (/ (* a b) 4.0)) c))