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

Percentage Accurate: 100.0% → 100.0%

Time: 6.7s

Alternatives: 10

Speedup: 1.2×

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, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. associate-+l+ 100.0%

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

   4. associate-/l* 99.9%

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

   5. distribute-frac-neg 99.9%

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

   6. associate-/r/ 100.0%

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

   7. *-commutative 100.0%

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

   8. +-commutative 100.0%

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

   9. fma-def 100.0%

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

   10. neg-mul-1 100.0%

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

   11. associate-/l* 100.0%

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

   12. associate-/r/ 100.0%

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

   13. *-commutative 100.0%

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

   14. metadata-eval 100.0%

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

   15. +-commutative 100.0%

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

   16. fma-def 100.0%

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

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 2: 100.0% accurate, 0.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. associate-/l* 99.9%

      \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
4. Add Preprocessing

5. Taylor expanded in x around 0 100.0%

   \[\leadsto \color{blue}{\left(-0.5 \cdot \left(y \cdot z\right) + 0.125 \cdot x\right)} + t \]
6. Step-by-step derivation

   1. *-commutative 100.0%

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

   2. associate-*l* 100.0%

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

   3. metadata-eval 100.0%

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

   4. distribute-rgt-neg-in 100.0%

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

   5. fma-udef 100.0%

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

   6. distribute-rgt-neg-in 100.0%

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

   7. metadata-eval 100.0%

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

7. Simplified 100.0%

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

8. Final simplification 100.0%

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

Alternative 3: 48.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(z \cdot -0.5\right)\\ \mathbf{if}\;y \leq -1.55 \cdot 10^{+119}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{+56}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{+15}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -1.18 \cdot 10^{-68}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-97}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -5.1 \cdot 10^{-191}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-102}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (* z -0.5))))
   (if (<= y -1.55e+119)
     t_1
     (if (<= y -6.8e+56)
       t
       (if (<= y -3.6e+33)
         t_1
         (if (<= y -1.05e+15)
           t
           (if (<= y -1.18e-68)
             (* 0.125 x)
             (if (<= y -7.8e-97)
               t
               (if (<= y -5.1e-191)
                 (* 0.125 x)
                 (if (<= y 1.85e-102) 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 <= -1.55e+119) {
		tmp = t_1;
	} else if (y <= -6.8e+56) {
		tmp = t;
	} else if (y <= -3.6e+33) {
		tmp = t_1;
	} else if (y <= -1.05e+15) {
		tmp = t;
	} else if (y <= -1.18e-68) {
		tmp = 0.125 * x;
	} else if (y <= -7.8e-97) {
		tmp = t;
	} else if (y <= -5.1e-191) {
		tmp = 0.125 * x;
	} else if (y <= 1.85e-102) {
		tmp = t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = y * (z * (-0.5d0))
    if (y <= (-1.55d+119)) then
        tmp = t_1
    else if (y <= (-6.8d+56)) then
        tmp = t
    else if (y <= (-3.6d+33)) then
        tmp = t_1
    else if (y <= (-1.05d+15)) then
        tmp = t
    else if (y <= (-1.18d-68)) then
        tmp = 0.125d0 * x
    else if (y <= (-7.8d-97)) then
        tmp = t
    else if (y <= (-5.1d-191)) then
        tmp = 0.125d0 * x
    else if (y <= 1.85d-102) then
        tmp = t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = y * (z * -0.5);
	double tmp;
	if (y <= -1.55e+119) {
		tmp = t_1;
	} else if (y <= -6.8e+56) {
		tmp = t;
	} else if (y <= -3.6e+33) {
		tmp = t_1;
	} else if (y <= -1.05e+15) {
		tmp = t;
	} else if (y <= -1.18e-68) {
		tmp = 0.125 * x;
	} else if (y <= -7.8e-97) {
		tmp = t;
	} else if (y <= -5.1e-191) {
		tmp = 0.125 * x;
	} else if (y <= 1.85e-102) {
		tmp = t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y * (z * -0.5)
	tmp = 0
	if y <= -1.55e+119:
		tmp = t_1
	elif y <= -6.8e+56:
		tmp = t
	elif y <= -3.6e+33:
		tmp = t_1
	elif y <= -1.05e+15:
		tmp = t
	elif y <= -1.18e-68:
		tmp = 0.125 * x
	elif y <= -7.8e-97:
		tmp = t
	elif y <= -5.1e-191:
		tmp = 0.125 * x
	elif y <= 1.85e-102:
		tmp = t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(z * -0.5))
	tmp = 0.0
	if (y <= -1.55e+119)
		tmp = t_1;
	elseif (y <= -6.8e+56)
		tmp = t;
	elseif (y <= -3.6e+33)
		tmp = t_1;
	elseif (y <= -1.05e+15)
		tmp = t;
	elseif (y <= -1.18e-68)
		tmp = Float64(0.125 * x);
	elseif (y <= -7.8e-97)
		tmp = t;
	elseif (y <= -5.1e-191)
		tmp = Float64(0.125 * x);
	elseif (y <= 1.85e-102)
		tmp = t;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y * (z * -0.5);
	tmp = 0.0;
	if (y <= -1.55e+119)
		tmp = t_1;
	elseif (y <= -6.8e+56)
		tmp = t;
	elseif (y <= -3.6e+33)
		tmp = t_1;
	elseif (y <= -1.05e+15)
		tmp = t;
	elseif (y <= -1.18e-68)
		tmp = 0.125 * x;
	elseif (y <= -7.8e-97)
		tmp = t;
	elseif (y <= -5.1e-191)
		tmp = 0.125 * x;
	elseif (y <= 1.85e-102)
		tmp = t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.55e+119], t$95$1, If[LessEqual[y, -6.8e+56], t, If[LessEqual[y, -3.6e+33], t$95$1, If[LessEqual[y, -1.05e+15], t, If[LessEqual[y, -1.18e-68], N[(0.125 * x), $MachinePrecision], If[LessEqual[y, -7.8e-97], t, If[LessEqual[y, -5.1e-191], N[(0.125 * x), $MachinePrecision], If[LessEqual[y, 1.85e-102], t, t$95$1]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.54999999999999998e119 or -6.80000000000000002e56 < y < -3.6000000000000003e33 or 1.8499999999999999e-102 < y

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.8%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(y \cdot z\right) + 0.125 \cdot x\right)} + t \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 56.9%

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

      1. *-commutative 56.9%

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

      2. associate-*r* 56.9%

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

      3. *-commutative 56.9%

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

   10. Simplified 56.9%

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

   if -1.54999999999999998e119 < y < -6.80000000000000002e56 or -3.6000000000000003e33 < y < -1.05e15 or -1.18000000000000005e-68 < y < -7.7999999999999997e-97 or -5.1000000000000002e-191 < y < 1.8499999999999999e-102

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.9%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 56.6%

      \[\leadsto \color{blue}{t} \]

   if -1.05e15 < y < -1.18000000000000005e-68 or -7.7999999999999997e-97 < y < -5.1000000000000002e-191

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 100.0%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(y \cdot z\right) + 0.125 \cdot x\right)} + t \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 55.4%

      \[\leadsto \color{blue}{0.125 \cdot x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+119}:\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{elif}\;y \leq -6.8 \cdot 10^{+56}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{+33}:\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{elif}\;y \leq -1.05 \cdot 10^{+15}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -1.18 \cdot 10^{-68}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;y \leq -7.8 \cdot 10^{-97}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -5.1 \cdot 10^{-191}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-102}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 69.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+119} \lor \neg \left(y \leq -2.5 \cdot 10^{+52} \lor \neg \left(y \leq -2.9 \cdot 10^{+34}\right) \land y \leq 1.5 \cdot 10^{-66}\right):\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.8e+119)
         (not (or (<= y -2.5e+52) (and (not (<= y -2.9e+34)) (<= y 1.5e-66)))))
   (* y (* z -0.5))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.8e+119) || !((y <= -2.5e+52) || (!(y <= -2.9e+34) && (y <= 1.5e-66)))) {
		tmp = y * (z * -0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

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
    real(8) :: tmp
    if ((y <= (-1.8d+119)) .or. (.not. (y <= (-2.5d+52)) .or. (.not. (y <= (-2.9d+34))) .and. (y <= 1.5d-66))) then
        tmp = y * (z * (-0.5d0))
    else
        tmp = t + (0.125d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.8e+119) || !((y <= -2.5e+52) || (!(y <= -2.9e+34) && (y <= 1.5e-66)))) {
		tmp = y * (z * -0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.8e+119) or not ((y <= -2.5e+52) or (not (y <= -2.9e+34) and (y <= 1.5e-66))):
		tmp = y * (z * -0.5)
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.8e+119) || !((y <= -2.5e+52) || (!(y <= -2.9e+34) && (y <= 1.5e-66))))
		tmp = Float64(y * Float64(z * -0.5));
	else
		tmp = Float64(t + Float64(0.125 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.8e+119) || ~(((y <= -2.5e+52) || (~((y <= -2.9e+34)) && (y <= 1.5e-66)))))
		tmp = y * (z * -0.5);
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.8e+119], N[Not[Or[LessEqual[y, -2.5e+52], And[N[Not[LessEqual[y, -2.9e+34]], $MachinePrecision], LessEqual[y, 1.5e-66]]]], $MachinePrecision]], N[(y * N[(z * -0.5), $MachinePrecision]), $MachinePrecision], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.80000000000000001e119 or -2.5e52 < y < -2.9000000000000001e34 or 1.5000000000000001e-66 < y

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.8%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(y \cdot z\right) + 0.125 \cdot x\right)} + t \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in y around inf 58.8%

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

      1. *-commutative 58.8%

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

      2. associate-*r* 58.8%

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

      3. *-commutative 58.8%

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

   10. Simplified 58.8%

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

   if -1.80000000000000001e119 < y < -2.5e52 or -2.9000000000000001e34 < y < 1.5000000000000001e-66

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.9%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 83.0%

      \[\leadsto \color{blue}{0.125 \cdot x} + t \]
3. Recombined 2 regimes into one program.

4. Final simplification 70.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{+119} \lor \neg \left(y \leq -2.5 \cdot 10^{+52} \lor \neg \left(y \leq -2.9 \cdot 10^{+34}\right) \land y \leq 1.5 \cdot 10^{-66}\right):\\ \;\;\;\;y \cdot \left(z \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -5 \cdot 10^{+53} \lor \neg \left(z \cdot y \leq 2 \cdot 10^{+69}\right):\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z y) -5e+53) (not (<= (* z y) 2e+69)))
   (- t (* (* z y) 0.5))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -5e+53) || !((z * y) <= 2e+69)) {
		tmp = t - ((z * y) * 0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

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
    real(8) :: tmp
    if (((z * y) <= (-5d+53)) .or. (.not. ((z * y) <= 2d+69))) then
        tmp = t - ((z * y) * 0.5d0)
    else
        tmp = t + (0.125d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -5e+53) || !((z * y) <= 2e+69)) {
		tmp = t - ((z * y) * 0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * y) <= -5e+53) or not ((z * y) <= 2e+69):
		tmp = t - ((z * y) * 0.5)
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * y) <= -5e+53) || !(Float64(z * y) <= 2e+69))
		tmp = Float64(t - Float64(Float64(z * y) * 0.5));
	else
		tmp = Float64(t + Float64(0.125 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * y) <= -5e+53) || ~(((z * y) <= 2e+69)))
		tmp = t - ((z * y) * 0.5);
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * y), $MachinePrecision], -5e+53], N[Not[LessEqual[N[(z * y), $MachinePrecision], 2e+69]], $MachinePrecision]], N[(t - N[(N[(z * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -5.0000000000000004e53 or 2.0000000000000001e69 < (*.f64 y z)

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.8%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 89.6%

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

   if -5.0000000000000004e53 < (*.f64 y z) < 2.0000000000000001e69

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.9%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 89.6%

      \[\leadsto \color{blue}{0.125 \cdot x} + t \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -5 \cdot 10^{+53} \lor \neg \left(z \cdot y \leq 2 \cdot 10^{+69}\right):\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot 0.5\\ \mathbf{if}\;t \leq -6 \cdot 10^{-62} \lor \neg \left(t \leq 8 \cdot 10^{+46}\right):\\ \;\;\;\;t - t\_1\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* z y) 0.5)))
   (if (or (<= t -6e-62) (not (<= t 8e+46))) (- t t_1) (- (* 0.125 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * y) * 0.5;
	double tmp;
	if ((t <= -6e-62) || !(t <= 8e+46)) {
		tmp = t - t_1;
	} else {
		tmp = (0.125 * x) - t_1;
	}
	return tmp;
}

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
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (z * y) * 0.5d0
    if ((t <= (-6d-62)) .or. (.not. (t <= 8d+46))) then
        tmp = t - t_1
    else
        tmp = (0.125d0 * x) - t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * y) * 0.5;
	double tmp;
	if ((t <= -6e-62) || !(t <= 8e+46)) {
		tmp = t - t_1;
	} else {
		tmp = (0.125 * x) - t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * y) * 0.5
	tmp = 0
	if (t <= -6e-62) or not (t <= 8e+46):
		tmp = t - t_1
	else:
		tmp = (0.125 * x) - t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * y) * 0.5)
	tmp = 0.0
	if ((t <= -6e-62) || !(t <= 8e+46))
		tmp = Float64(t - t_1);
	else
		tmp = Float64(Float64(0.125 * x) - t_1);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * y) * 0.5;
	tmp = 0.0;
	if ((t <= -6e-62) || ~((t <= 8e+46)))
		tmp = t - t_1;
	else
		tmp = (0.125 * x) - t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * y), $MachinePrecision] * 0.5), $MachinePrecision]}, If[Or[LessEqual[t, -6e-62], N[Not[LessEqual[t, 8e+46]], $MachinePrecision]], N[(t - t$95$1), $MachinePrecision], N[(N[(0.125 * x), $MachinePrecision] - t$95$1), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if t < -6.0000000000000002e-62 or 7.9999999999999999e46 < t

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.9%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 86.9%

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

   if -6.0000000000000002e-62 < t < 7.9999999999999999e46

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.8%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in t around 0 93.3%

      \[\leadsto \color{blue}{0.125 \cdot x - 0.5 \cdot \left(y \cdot z\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6 \cdot 10^{-62} \lor \neg \left(t \leq 8 \cdot 10^{+46}\right):\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x - \left(z \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 49.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-62}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+45}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -5.5e-62) t (if (<= t 9.5e+45) (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.5e-62) {
		tmp = t;
	} else if (t <= 9.5e+45) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

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
    real(8) :: tmp
    if (t <= (-5.5d-62)) then
        tmp = t
    else if (t <= 9.5d+45) then
        tmp = 0.125d0 * x
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -5.5e-62) {
		tmp = t;
	} else if (t <= 9.5e+45) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -5.5e-62:
		tmp = t
	elif t <= 9.5e+45:
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -5.5e-62)
		tmp = t;
	elseif (t <= 9.5e+45)
		tmp = Float64(0.125 * x);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -5.5e-62)
		tmp = t;
	elseif (t <= 9.5e+45)
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -5.5e-62], t, If[LessEqual[t, 9.5e+45], N[(0.125 * x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if t < -5.50000000000000022e-62 or 9.4999999999999998e45 < t

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.9%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in t around inf 60.9%

      \[\leadsto \color{blue}{t} \]

   if -5.50000000000000022e-62 < t < 9.4999999999999998e45

   1. Initial program 100.0%

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

      1. metadata-eval 100.0%

         \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

      2. associate-/l* 99.8%

         \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

   3. Simplified 99.8%

      \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 100.0%

      \[\leadsto \color{blue}{\left(-0.5 \cdot \left(y \cdot z\right) + 0.125 \cdot x\right)} + t \]
   6. Step-by-step derivation

      1. *-commutative 100.0%

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

      2. associate-*l* 100.0%

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

      3. metadata-eval 100.0%

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

      4. distribute-rgt-neg-in 100.0%

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

      5. fma-udef 100.0%

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

      6. distribute-rgt-neg-in 100.0%

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

      7. metadata-eval 100.0%

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

   7. Simplified 100.0%

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

   8. Taylor expanded in x around inf 41.1%

      \[\leadsto \color{blue}{0.125 \cdot x} \]
3. Recombined 2 regimes into one program.

4. Final simplification 51.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.5 \cdot 10^{-62}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+45}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 99.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ t + \left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ t (- (* 0.125 x) (/ y (/ 2.0 z)))))

C

double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - (y / (2.0 / z)));
}

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 = t + ((0.125d0 * x) - (y / (2.0d0 / z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - (y / (2.0 / z)));
}

Python

def code(x, y, z, t):
	return t + ((0.125 * x) - (y / (2.0 / z)))

Julia

function code(x, y, z, t)
	return Float64(t + Float64(Float64(0.125 * x) - Float64(y / Float64(2.0 / z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t + ((0.125 * x) - (y / (2.0 / z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(t + N[(N[(0.125 * x), $MachinePrecision] - N[(y / N[(2.0 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 100.0%

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. associate-/l* 99.9%

      \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
4. Add Preprocessing

5. Final simplification 99.9%

   \[\leadsto t + \left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) \]
6. Add Preprocessing

Alternative 9: 100.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ t + \left(0.125 \cdot x - \frac{z \cdot y}{2}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (+ t (- (* 0.125 x) (/ (* z y) 2.0))))

C

double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - ((z * y) / 2.0));
}

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 = t + ((0.125d0 * x) - ((z * y) / 2.0d0))
end function

Java

public static double code(double x, double y, double z, double t) {
	return t + ((0.125 * x) - ((z * y) / 2.0));
}

Python

def code(x, y, z, t):
	return t + ((0.125 * x) - ((z * y) / 2.0))

Julia

function code(x, y, z, t)
	return Float64(t + Float64(Float64(0.125 * x) - Float64(Float64(z * y) / 2.0)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = t + ((0.125 * x) - ((z * y) / 2.0));
end

Wolfram

code[x_, y_, z_, t_] := N[(t + N[(N[(0.125 * x), $MachinePrecision] - N[(N[(z * y), $MachinePrecision] / 2.0), $MachinePrecision]), $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. Final simplification 100.0%

   \[\leadsto t + \left(0.125 \cdot x - \frac{z \cdot y}{2}\right) \]
4. Add Preprocessing

Alternative 10: 33.5% accurate, 13.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 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 = t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := t

Derivation

1. Initial program 100.0%

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

   1. metadata-eval 100.0%

      \[\leadsto \left(\color{blue}{0.125} \cdot x - \frac{y \cdot z}{2}\right) + t \]

   2. associate-/l* 99.9%

      \[\leadsto \left(0.125 \cdot x - \color{blue}{\frac{y}{\frac{2}{z}}}\right) + t \]

3. Simplified 99.9%

   \[\leadsto \color{blue}{\left(0.125 \cdot x - \frac{y}{\frac{2}{z}}\right) + t} \]
4. Add Preprocessing

5. Taylor expanded in t around inf 36.7%

   \[\leadsto \color{blue}{t} \]

6. Final simplification 36.7%

   \[\leadsto t \]
7. Add Preprocessing

Developer target: 100.0% accurate, 1.2× 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 2024027 
(FPCore (x y z t)
  :name "Diagrams.Solve.Polynomial:quartForm  from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (- (+ (/ x 8.0) t) (* (/ z 2.0) y))

  (+ (- (* (/ 1.0 8.0) x) (/ (* y z) 2.0)) t))