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

Percentage Accurate: 97.7% → 99.1%

Time: 11.8s

Alternatives: 14

Speedup: 1.6×

• 33.2% 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: 97.7% 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: 99.1% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.2%

   \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
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. lift-+.f64 N/A

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

   4. associate--l+ N/A

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

   5. associate-+l+ N/A

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

   6. lift-*.f64 N/A

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

   7. *-commutative N/A

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

   8. lower-fma.f64 N/A

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

   9. lower-+.f64 N/A

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

4. Applied rewrites 98.8%

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

   1. lift-+.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. associate-*r* N/A

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

   6. lift-*.f64 N/A

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

   7. lift-*.f64 N/A

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

   8. +-commutative N/A

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

   9. associate-+l+ N/A

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

   10. lift-*.f64 N/A

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

   11. *-commutative N/A

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

   12. lower-fma.f64 N/A

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

   13. lower-fma.f64 99.2

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

6. Applied rewrites 99.2%

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

Alternative 2: 76.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, x, 0.0625 \cdot \left(z \cdot t\right)\right)\\ t_2 := y \cdot x + \frac{z \cdot t}{16}\\ t_3 := \mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+189}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -2 \cdot 10^{+35}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+277}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a b c)
 :precision binary64
 (let* ((t_1 (fma y x (* 0.0625 (* z t))))
        (t_2 (+ (* y x) (/ (* z t) 16.0)))
        (t_3 (fma y x (* -0.25 (* b a)))))
   (if (<= t_2 -5e+189)
     t_1
     (if (<= t_2 -2e+35)
       t_3
       (if (<= t_2 1e+114)
         (fma -0.25 (* b a) c)
         (if (<= t_2 5e+277) t_3 t_1))))))

C

double code(double x, double y, double z, double t, double a, double b, double c) {
	double t_1 = fma(y, x, (0.0625 * (z * t)));
	double t_2 = (y * x) + ((z * t) / 16.0);
	double t_3 = fma(y, x, (-0.25 * (b * a)));
	double tmp;
	if (t_2 <= -5e+189) {
		tmp = t_1;
	} else if (t_2 <= -2e+35) {
		tmp = t_3;
	} else if (t_2 <= 1e+114) {
		tmp = fma(-0.25, (b * a), c);
	} else if (t_2 <= 5e+277) {
		tmp = t_3;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a, b, c)
	t_1 = fma(y, x, Float64(0.0625 * Float64(z * t)))
	t_2 = Float64(Float64(y * x) + Float64(Float64(z * t) / 16.0))
	t_3 = fma(y, x, Float64(-0.25 * Float64(b * a)))
	tmp = 0.0
	if (t_2 <= -5e+189)
		tmp = t_1;
	elseif (t_2 <= -2e+35)
		tmp = t_3;
	elseif (t_2 <= 1e+114)
		tmp = fma(-0.25, Float64(b * a), c);
	elseif (t_2 <= 5e+277)
		tmp = t_3;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_, b_, c_] := Block[{t$95$1 = N[(y * x + N[(0.0625 * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y * x), $MachinePrecision] + N[(N[(z * t), $MachinePrecision] / 16.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(y * x + N[(-0.25 * N[(b * a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+189], t$95$1, If[LessEqual[t$95$2, -2e+35], t$95$3, If[LessEqual[t$95$2, 1e+114], N[(-0.25 * N[(b * a), $MachinePrecision] + c), $MachinePrecision], If[LessEqual[t$95$2, 5e+277], t$95$3, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -5.0000000000000004e189 or 4.99999999999999982e277 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64)))

   1. Initial program 93.6%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in z around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 89.2

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

   7. Applied rewrites 89.2%

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

   if -5.0000000000000004e189 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < -1.9999999999999999e35 or 1e114 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 4.99999999999999982e277

   1. Initial program 99.8%

      \[\left(\left(x \cdot y + \frac{z \cdot t}{16}\right) - \frac{a \cdot b}{4}\right) + c \]
   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. lift-+.f64 N/A

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

      4. associate--l+ N/A

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

      5. associate-+l+ N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-+.f64 N/A

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in a around inf

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

      1. lower-*.f64 N/A

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

      2. lower-*.f64 56.2

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

   7. Applied rewrites 56.2%

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

   if -1.9999999999999999e35 < (+.f64 (*.f64 x y) (/.f64 (*.f64 z t) #s(literal 16 binary64))) < 1e114

   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 90.5

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

   5. Applied rewrites 90.5%

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

   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 80.2%

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

4. Final simplification 76.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot x + \frac{z \cdot t}{16} \leq -5 \cdot 10^{+189}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(z \cdot t\right)\right)\\ \mathbf{elif}\;y \cdot x + \frac{z \cdot t}{16} \leq -2 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\\ \mathbf{elif}\;y \cdot x + \frac{z \cdot t}{16} \leq 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(-0.25, b \cdot a, c\right)\\ \mathbf{elif}\;y \cdot x + \frac{z \cdot t}{16} \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\mathsf{fma}\left(y, x, -0.25 \cdot \left(b \cdot a\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, x, 0.0625 \cdot \left(z \cdot t\right)\right)\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024223 
(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))