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

Percentage Accurate: 100.0% → 100.0%

Time: 5.3s

Alternatives: 6

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, 1.2× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (0.125 * x) + (t + ((y * z) / -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 = (0.125d0 * x) + (t + ((y * z) / (-2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(0.125 * x), $MachinePrecision] + N[(t + N[(N[(y * z), $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. Step-by-step derivation

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

3. Simplified 100.0%

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

Alternative 2: 56.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot z\right) \cdot -0.5\\ \mathbf{if}\;y \cdot z \leq -1.45 \cdot 10^{+55}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot z \leq 5.3 \cdot 10^{-123}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \cdot z \leq 2.2 \cdot 10^{-67}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* y z) -0.5)))
   (if (<= (* y z) -1.45e+55)
     t_1
     (if (<= (* y z) 5.3e-123) t (if (<= (* y z) 2.2e-67) (* 0.125 x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) * -0.5;
	double tmp;
	if ((y * z) <= -1.45e+55) {
		tmp = t_1;
	} else if ((y * z) <= 5.3e-123) {
		tmp = t;
	} else if ((y * z) <= 2.2e-67) {
		tmp = 0.125 * x;
	} 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 * z) <= (-1.45d+55)) then
        tmp = t_1
    else if ((y * z) <= 5.3d-123) then
        tmp = t
    else if ((y * z) <= 2.2d-67) then
        tmp = 0.125d0 * x
    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 * z) <= -1.45e+55) {
		tmp = t_1;
	} else if ((y * z) <= 5.3e-123) {
		tmp = t;
	} else if ((y * z) <= 2.2e-67) {
		tmp = 0.125 * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) * -0.5
	tmp = 0
	if (y * z) <= -1.45e+55:
		tmp = t_1
	elif (y * z) <= 5.3e-123:
		tmp = t
	elif (y * z) <= 2.2e-67:
		tmp = 0.125 * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) * -0.5)
	tmp = 0.0
	if (Float64(y * z) <= -1.45e+55)
		tmp = t_1;
	elseif (Float64(y * z) <= 5.3e-123)
		tmp = t;
	elseif (Float64(y * z) <= 2.2e-67)
		tmp = Float64(0.125 * x);
	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 * z) <= -1.45e+55)
		tmp = t_1;
	elseif ((y * z) <= 5.3e-123)
		tmp = t;
	elseif ((y * z) <= 2.2e-67)
		tmp = 0.125 * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -1.45e+55], t$95$1, If[LessEqual[N[(y * z), $MachinePrecision], 5.3e-123], t, If[LessEqual[N[(y * z), $MachinePrecision], 2.2e-67], N[(0.125 * x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -1.4499999999999999e55 or 2.2000000000000001e-67 < (*.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. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-lowering-*.f64 73.4%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 73.4%

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

   if -1.4499999999999999e55 < (*.f64 y z) < 5.29999999999999971e-123

   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 N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t} \]
   6. Step-by-step derivation

   7. Simplified 54.9%

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

   if 5.29999999999999971e-123 < (*.f64 y z) < 2.2000000000000001e-67

   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 N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 72.2%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{8}, \color{blue}{x}\right) \]

   7. Simplified 72.2%

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

4. Final simplification 65.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -1.45 \cdot 10^{+55}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -0.5\\ \mathbf{elif}\;y \cdot z \leq 5.3 \cdot 10^{-123}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \cdot z \leq 2.2 \cdot 10^{-67}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(y \cdot z\right) \cdot -0.5\\ \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot z \leq 10^{-56}:\\ \;\;\;\;t - x \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ t (* (* y z) -0.5))))
   (if (<= (* y z) -5e+95)
     t_1
     (if (<= (* y z) 1e-56) (- t (* x -0.125)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t + ((y * z) * -0.5);
	double tmp;
	if ((y * z) <= -5e+95) {
		tmp = t_1;
	} else if ((y * z) <= 1e-56) {
		tmp = t - (x * -0.125);
	} 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 = t + ((y * z) * (-0.5d0))
    if ((y * z) <= (-5d+95)) then
        tmp = t_1
    else if ((y * z) <= 1d-56) then
        tmp = t - (x * (-0.125d0))
    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 = t + ((y * z) * -0.5);
	double tmp;
	if ((y * z) <= -5e+95) {
		tmp = t_1;
	} else if ((y * z) <= 1e-56) {
		tmp = t - (x * -0.125);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = t + ((y * z) * -0.5)
	tmp = 0
	if (y * z) <= -5e+95:
		tmp = t_1
	elif (y * z) <= 1e-56:
		tmp = t - (x * -0.125)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(t + Float64(Float64(y * z) * -0.5))
	tmp = 0.0
	if (Float64(y * z) <= -5e+95)
		tmp = t_1;
	elseif (Float64(y * z) <= 1e-56)
		tmp = Float64(t - Float64(x * -0.125));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = t + ((y * z) * -0.5);
	tmp = 0.0;
	if ((y * z) <= -5e+95)
		tmp = t_1;
	elseif ((y * z) <= 1e-56)
		tmp = t - (x * -0.125);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t + N[(N[(y * z), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -5e+95], t$95$1, If[LessEqual[N[(y * z), $MachinePrecision], 1e-56], N[(t - N[(x * -0.125), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -5.00000000000000025e95 or 1e-56 < (*.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. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0

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

      1. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(t, \color{blue}{\left(\frac{-1}{2} \cdot \left(y \cdot z\right)\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(y \cdot z\right)}\right)\right) \]

      3. *-lowering-*.f64 89.9%

         \[\leadsto \mathsf{+.f64}\left(t, \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right)\right) \]

   7. Simplified 89.9%

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

   if -5.00000000000000025e95 < (*.f64 y z) < 1e-56

   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 N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

         \[\leadsto t + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot x\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto t + \left(\mathsf{neg}\left(-1 \cdot \left(\frac{1}{8} \cdot x\right)\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(t, \color{blue}{\left(-1 \cdot \left(\frac{1}{8} \cdot x\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(t, \left(\mathsf{neg}\left(\frac{1}{8} \cdot x\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(t, \left(\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(t, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}\right)\right) \]

      9. metadata-eval 94.0%

         \[\leadsto \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(x, \frac{-1}{8}\right)\right) \]

   7. Simplified 94.0%

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

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+95}:\\ \;\;\;\;t + \left(y \cdot z\right) \cdot -0.5\\ \mathbf{elif}\;y \cdot z \leq 10^{-56}:\\ \;\;\;\;t - x \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t + \left(y \cdot z\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot z\right) \cdot -0.5\\ \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+95}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+91}:\\ \;\;\;\;t - x \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* y z) -0.5)))
   (if (<= (* y z) -5e+95)
     t_1
     (if (<= (* y z) 5e+91) (- t (* x -0.125)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) * -0.5;
	double tmp;
	if ((y * z) <= -5e+95) {
		tmp = t_1;
	} else if ((y * z) <= 5e+91) {
		tmp = t - (x * -0.125);
	} 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 * z) <= (-5d+95)) then
        tmp = t_1
    else if ((y * z) <= 5d+91) then
        tmp = t - (x * (-0.125d0))
    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 * z) <= -5e+95) {
		tmp = t_1;
	} else if ((y * z) <= 5e+91) {
		tmp = t - (x * -0.125);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) * -0.5
	tmp = 0
	if (y * z) <= -5e+95:
		tmp = t_1
	elif (y * z) <= 5e+91:
		tmp = t - (x * -0.125)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) * -0.5)
	tmp = 0.0
	if (Float64(y * z) <= -5e+95)
		tmp = t_1;
	elseif (Float64(y * z) <= 5e+91)
		tmp = Float64(t - Float64(x * -0.125));
	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 * z) <= -5e+95)
		tmp = t_1;
	elseif ((y * z) <= 5e+91)
		tmp = t - (x * -0.125);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] * -0.5), $MachinePrecision]}, If[LessEqual[N[(y * z), $MachinePrecision], -5e+95], t$95$1, If[LessEqual[N[(y * z), $MachinePrecision], 5e+91], N[(t - N[(x * -0.125), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -5.00000000000000025e95 or 5.0000000000000002e91 < (*.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. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \color{blue}{\left(y \cdot z\right)}\right) \]

      2. *-lowering-*.f64 85.6%

         \[\leadsto \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{*.f64}\left(y, \color{blue}{z}\right)\right) \]

   7. Simplified 85.6%

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

   if -5.00000000000000025e95 < (*.f64 y z) < 5.0000000000000002e91

   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 N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0

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

      1. remove-double-neg N/A

         \[\leadsto t + \left(\mathsf{neg}\left(\left(\mathsf{neg}\left(\frac{1}{8} \cdot x\right)\right)\right)\right) \]

      2. mul-1-neg N/A

         \[\leadsto t + \left(\mathsf{neg}\left(-1 \cdot \left(\frac{1}{8} \cdot x\right)\right)\right) \]

      3. unsub-neg N/A

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

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(t, \color{blue}{\left(-1 \cdot \left(\frac{1}{8} \cdot x\right)\right)}\right) \]

      5. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(t, \left(\mathsf{neg}\left(\frac{1}{8} \cdot x\right)\right)\right) \]

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

         \[\leadsto \mathsf{\_.f64}\left(t, \left(\left(\mathsf{neg}\left(\frac{1}{8}\right)\right) \cdot \color{blue}{x}\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(t, \left(x \cdot \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(x, \color{blue}{\left(\mathsf{neg}\left(\frac{1}{8}\right)\right)}\right)\right) \]

      9. metadata-eval 88.6%

         \[\leadsto \mathsf{\_.f64}\left(t, \mathsf{*.f64}\left(x, \frac{-1}{8}\right)\right) \]

   7. Simplified 88.6%

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

4. Final simplification 87.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot z \leq -5 \cdot 10^{+95}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -0.5\\ \mathbf{elif}\;y \cdot z \leq 5 \cdot 10^{+91}:\\ \;\;\;\;t - x \cdot -0.125\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot z\right) \cdot -0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 49.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{+81}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-10}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.8e+81) t (if (<= t 3.2e-10) (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.8e+81) {
		tmp = t;
	} else if (t <= 3.2e-10) {
		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 <= (-2.8d+81)) then
        tmp = t
    else if (t <= 3.2d-10) 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 <= -2.8e+81) {
		tmp = t;
	} else if (t <= 3.2e-10) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.8e+81:
		tmp = t
	elif t <= 3.2e-10:
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.8e+81)
		tmp = t;
	elseif (t <= 3.2e-10)
		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 <= -2.8e+81)
		tmp = t;
	elseif (t <= 3.2e-10)
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.8e+81], t, If[LessEqual[t, 3.2e-10], N[(0.125 * x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if t < -2.79999999999999995e81 or 3.19999999999999981e-10 < 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. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf

      \[\leadsto \color{blue}{t} \]
   6. Step-by-step derivation

   7. Simplified 63.1%

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

   if -2.79999999999999995e81 < t < 3.19999999999999981e-10

   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 N/A

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

      2. associate-+l+ N/A

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

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

      4. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

      5. metadata-eval N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

      6. +-commutative N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      7. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

      8. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

      11. metadata-eval 100.0%

          \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf

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

      1. *-lowering-*.f64 43.3%

         \[\leadsto \mathsf{*.f64}\left(\frac{1}{8}, \color{blue}{x}\right) \]

   7. Simplified 43.3%

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

Alternative 6: 33.1% 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. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\left(\frac{1}{8} \cdot x\right), \color{blue}{\left(\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right) + t\right)}\right) \]

   4. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\left(\frac{1}{8}\right), x\right), \left(\color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)} + t\right)\right) \]

   5. metadata-eval N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(\left(\mathsf{neg}\left(\color{blue}{\frac{y \cdot z}{2}}\right)\right) + t\right)\right) \]

   6. +-commutative N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \left(t + \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

   7. +-lowering-+.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \color{blue}{\left(\mathsf{neg}\left(\frac{y \cdot z}{2}\right)\right)}\right)\right) \]

   8. distribute-neg-frac2 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \left(\frac{y \cdot z}{\color{blue}{\mathsf{neg}\left(2\right)}}\right)\right)\right) \]

   9. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\left(y \cdot z\right), \color{blue}{\left(\mathsf{neg}\left(2\right)\right)}\right)\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), \left(\mathsf{neg}\left(\color{blue}{2}\right)\right)\right)\right)\right) \]

   11. metadata-eval 100.0%

       \[\leadsto \mathsf{+.f64}\left(\mathsf{*.f64}\left(\frac{1}{8}, x\right), \mathsf{+.f64}\left(t, \mathsf{/.f64}\left(\mathsf{*.f64}\left(y, z\right), -2\right)\right)\right) \]

3. Simplified 100.0%

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

5. Taylor expanded in t around inf

   \[\leadsto \color{blue}{t} \]
6. Step-by-step derivation

7. Simplified 34.6%

   \[\leadsto \color{blue}{t} \]
8. Add Preprocessing

Developer Target 1: 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 2024145 
(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))