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

Percentage Accurate: 100.0% → 100.0%

Time: 5.9s

Alternatives: 8

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

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(z * -0.5), $MachinePrecision] * y + N[(0.125 * x), $MachinePrecision]), $MachinePrecision] + t), $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. associate-+l- 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-+l- 100.0%

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

   4. metadata-eval 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-/l* 100.0%

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

3. Simplified 100.0%

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

   1. sub-neg 100.0%

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

   2. +-commutative 100.0%

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

   3. *-commutative 100.0%

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

   4. distribute-lft-neg-in 100.0%

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

   5. div-inv 100.0%

      \[\leadsto \left(\left(-\color{blue}{z \cdot \frac{1}{2}}\right) \cdot y + 0.125 \cdot x\right) + t \]

   6. distribute-rgt-neg-in 100.0%

      \[\leadsto \left(\color{blue}{\left(z \cdot \left(-\frac{1}{2}\right)\right)} \cdot y + 0.125 \cdot x\right) + t \]

   7. metadata-eval 100.0%

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

   8. metadata-eval 100.0%

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

   9. add-sqr-sqrt 51.9%

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

   10. sqrt-unprod 70.5%

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

   11. swap-sqr 70.5%

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

   12. metadata-eval 70.5%

       \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \color{blue}{0.25}} \cdot y + 0.125 \cdot x\right) + t \]

   13. metadata-eval 70.5%

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

   14. metadata-eval 70.5%

       \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot 0.5\right)} \cdot y + 0.125 \cdot x\right) + t \]

   15. metadata-eval 70.5%

       \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{2}}\right)} \cdot y + 0.125 \cdot x\right) + t \]

   16. swap-sqr 70.5%

       \[\leadsto \left(\sqrt{\color{blue}{\left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right)}} \cdot y + 0.125 \cdot x\right) + t \]

   17. div-inv 70.5%

       \[\leadsto \left(\sqrt{\color{blue}{\frac{z}{2}} \cdot \left(z \cdot \frac{1}{2}\right)} \cdot y + 0.125 \cdot x\right) + t \]

   18. div-inv 70.5%

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

   19. sqrt-unprod 29.8%

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

   20. add-sqr-sqrt 64.0%

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

6. Applied egg-rr 100.0%

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

7. Final simplification 100.0%

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

Alternative 2: 87.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(z \cdot y\right) \cdot 0.5\\ \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+59}:\\ \;\;\;\;t - t\_1\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{+101}:\\ \;\;\;\;0.125 \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x - t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* z y) 0.5)))
   (if (<= (* z y) -1e+59)
     (- t t_1)
     (if (<= (* z y) 5e+101) (+ (* 0.125 x) t) (- (* 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 ((z * y) <= -1e+59) {
		tmp = t - t_1;
	} else if ((z * y) <= 5e+101) {
		tmp = (0.125 * x) + t;
	} 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 ((z * y) <= (-1d+59)) then
        tmp = t - t_1
    else if ((z * y) <= 5d+101) then
        tmp = (0.125d0 * x) + t
    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 ((z * y) <= -1e+59) {
		tmp = t - t_1;
	} else if ((z * y) <= 5e+101) {
		tmp = (0.125 * x) + t;
	} 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 (z * y) <= -1e+59:
		tmp = t - t_1
	elif (z * y) <= 5e+101:
		tmp = (0.125 * x) + t
	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 (Float64(z * y) <= -1e+59)
		tmp = Float64(t - t_1);
	elseif (Float64(z * y) <= 5e+101)
		tmp = Float64(Float64(0.125 * x) + t);
	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 ((z * y) <= -1e+59)
		tmp = t - t_1;
	elseif ((z * y) <= 5e+101)
		tmp = (0.125 * x) + t;
	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[LessEqual[N[(z * y), $MachinePrecision], -1e+59], N[(t - t$95$1), $MachinePrecision], If[LessEqual[N[(z * y), $MachinePrecision], 5e+101], N[(N[(0.125 * x), $MachinePrecision] + t), $MachinePrecision], N[(N[(0.125 * x), $MachinePrecision] - t$95$1), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 y z) < -9.99999999999999972e58

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 89.3%

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

   if -9.99999999999999972e58 < (*.f64 y z) < 4.99999999999999989e101

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 92.0%

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

   if 4.99999999999999989e101 < (*.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. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around 0 91.6%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+59}:\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{elif}\;z \cdot y \leq 5 \cdot 10^{+101}:\\ \;\;\;\;0.125 \cdot x + t\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x - \left(z \cdot y\right) \cdot 0.5\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+59} \lor \neg \left(z \cdot y \leq 10^{+155}\right):\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z y) -1e+59) (not (<= (* z y) 1e+155)))
   (- t (* (* z y) 0.5))
   (+ (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -1e+59) || !((z * y) <= 1e+155)) {
		tmp = t - ((z * y) * 0.5);
	} else {
		tmp = (0.125 * x) + 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 (((z * y) <= (-1d+59)) .or. (.not. ((z * y) <= 1d+155))) then
        tmp = t - ((z * y) * 0.5d0)
    else
        tmp = (0.125d0 * x) + t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -1e+59) || !((z * y) <= 1e+155)) {
		tmp = t - ((z * y) * 0.5);
	} else {
		tmp = (0.125 * x) + t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * y) <= -1e+59) or not ((z * y) <= 1e+155):
		tmp = t - ((z * y) * 0.5)
	else:
		tmp = (0.125 * x) + t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * y) <= -1e+59) || !(Float64(z * y) <= 1e+155))
		tmp = Float64(t - Float64(Float64(z * y) * 0.5));
	else
		tmp = Float64(Float64(0.125 * x) + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * y) <= -1e+59) || ~(((z * y) <= 1e+155)))
		tmp = t - ((z * y) * 0.5);
	else
		tmp = (0.125 * x) + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * y), $MachinePrecision], -1e+59], N[Not[LessEqual[N[(z * y), $MachinePrecision], 1e+155]], $MachinePrecision]], N[(t - N[(N[(z * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * x), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -9.99999999999999972e58 or 1.00000000000000001e155 < (*.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. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around 0 91.6%

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

   if -9.99999999999999972e58 < (*.f64 y z) < 1.00000000000000001e155

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 90.6%

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

4. Final simplification 91.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+59} \lor \neg \left(z \cdot y \leq 10^{+155}\right):\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + t\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 50.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+82}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-192}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot y\right)\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-54}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.25e+82)
   t
   (if (<= t 1.7e-192) (* z (* -0.5 y)) (if (<= t 9.2e-54) (* 0.125 x) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.25e+82) {
		tmp = t;
	} else if (t <= 1.7e-192) {
		tmp = z * (-0.5 * y);
	} else if (t <= 9.2e-54) {
		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 <= (-1.25d+82)) then
        tmp = t
    else if (t <= 1.7d-192) then
        tmp = z * ((-0.5d0) * y)
    else if (t <= 9.2d-54) 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 <= -1.25e+82) {
		tmp = t;
	} else if (t <= 1.7e-192) {
		tmp = z * (-0.5 * y);
	} else if (t <= 9.2e-54) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.25e+82:
		tmp = t
	elif t <= 1.7e-192:
		tmp = z * (-0.5 * y)
	elif t <= 9.2e-54:
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.25e+82)
		tmp = t;
	elseif (t <= 1.7e-192)
		tmp = Float64(z * Float64(-0.5 * y));
	elseif (t <= 9.2e-54)
		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 <= -1.25e+82)
		tmp = t;
	elseif (t <= 1.7e-192)
		tmp = z * (-0.5 * y);
	elseif (t <= 9.2e-54)
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.25e+82], t, If[LessEqual[t, 1.7e-192], N[(z * N[(-0.5 * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.2e-54], N[(0.125 * x), $MachinePrecision], t]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.25000000000000004e82 or 9.1999999999999996e-54 < 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. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 65.1%

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

   if -1.25000000000000004e82 < t < 1.70000000000000001e-192

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. div-inv 100.0%

         \[\leadsto \left(\left(-\color{blue}{z \cdot \frac{1}{2}}\right) \cdot y + 0.125 \cdot x\right) + t \]

      6. distribute-rgt-neg-in 100.0%

         \[\leadsto \left(\color{blue}{\left(z \cdot \left(-\frac{1}{2}\right)\right)} \cdot y + 0.125 \cdot x\right) + t \]

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. add-sqr-sqrt 45.7%

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

      10. sqrt-unprod 61.6%

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

      11. swap-sqr 61.6%

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

      12. metadata-eval 61.6%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \color{blue}{0.25}} \cdot y + 0.125 \cdot x\right) + t \]

      13. metadata-eval 61.6%

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

      14. metadata-eval 61.6%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot 0.5\right)} \cdot y + 0.125 \cdot x\right) + t \]

      15. metadata-eval 61.6%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{2}}\right)} \cdot y + 0.125 \cdot x\right) + t \]

      16. swap-sqr 61.6%

          \[\leadsto \left(\sqrt{\color{blue}{\left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right)}} \cdot y + 0.125 \cdot x\right) + t \]

      17. div-inv 61.6%

          \[\leadsto \left(\sqrt{\color{blue}{\frac{z}{2}} \cdot \left(z \cdot \frac{1}{2}\right)} \cdot y + 0.125 \cdot x\right) + t \]

      18. div-inv 61.6%

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

      19. sqrt-unprod 24.7%

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

      20. add-sqr-sqrt 42.9%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in z around inf 55.1%

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

      1. associate-*r* 55.1%

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

      2. *-commutative 55.1%

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

   9. Simplified 55.1%

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

   if 1.70000000000000001e-192 < t < 9.1999999999999996e-54

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. div-inv 100.0%

         \[\leadsto \left(\left(-\color{blue}{z \cdot \frac{1}{2}}\right) \cdot y + 0.125 \cdot x\right) + t \]

      6. distribute-rgt-neg-in 100.0%

         \[\leadsto \left(\color{blue}{\left(z \cdot \left(-\frac{1}{2}\right)\right)} \cdot y + 0.125 \cdot x\right) + t \]

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. add-sqr-sqrt 64.1%

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

      10. sqrt-unprod 79.2%

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

      11. swap-sqr 79.2%

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

      12. metadata-eval 79.2%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \color{blue}{0.25}} \cdot y + 0.125 \cdot x\right) + t \]

      13. metadata-eval 79.2%

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

      14. metadata-eval 79.2%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot 0.5\right)} \cdot y + 0.125 \cdot x\right) + t \]

      15. metadata-eval 79.2%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{2}}\right)} \cdot y + 0.125 \cdot x\right) + t \]

      16. swap-sqr 79.2%

          \[\leadsto \left(\sqrt{\color{blue}{\left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right)}} \cdot y + 0.125 \cdot x\right) + t \]

      17. div-inv 79.2%

          \[\leadsto \left(\sqrt{\color{blue}{\frac{z}{2}} \cdot \left(z \cdot \frac{1}{2}\right)} \cdot y + 0.125 \cdot x\right) + t \]

      18. div-inv 79.2%

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

      19. sqrt-unprod 25.0%

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

      20. add-sqr-sqrt 71.6%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 65.6%

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

4. Final simplification 61.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+82}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{-192}:\\ \;\;\;\;z \cdot \left(-0.5 \cdot y\right)\\ \mathbf{elif}\;t \leq 9.2 \cdot 10^{-54}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 68.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-80} \lor \neg \left(z \leq 1.1 \cdot 10^{+188}\right):\\ \;\;\;\;z \cdot \left(-0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -7.2e-80) (not (<= z 1.1e+188)))
   (* z (* -0.5 y))
   (+ (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.2e-80) || !(z <= 1.1e+188)) {
		tmp = z * (-0.5 * y);
	} else {
		tmp = (0.125 * x) + 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 ((z <= (-7.2d-80)) .or. (.not. (z <= 1.1d+188))) then
        tmp = z * ((-0.5d0) * y)
    else
        tmp = (0.125d0 * x) + t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -7.2e-80) || !(z <= 1.1e+188)) {
		tmp = z * (-0.5 * y);
	} else {
		tmp = (0.125 * x) + t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -7.2e-80) or not (z <= 1.1e+188):
		tmp = z * (-0.5 * y)
	else:
		tmp = (0.125 * x) + t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -7.2e-80) || !(z <= 1.1e+188))
		tmp = Float64(z * Float64(-0.5 * y));
	else
		tmp = Float64(Float64(0.125 * x) + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -7.2e-80) || ~((z <= 1.1e+188)))
		tmp = z * (-0.5 * y);
	else
		tmp = (0.125 * x) + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -7.2e-80], N[Not[LessEqual[z, 1.1e+188]], $MachinePrecision]], N[(z * N[(-0.5 * y), $MachinePrecision]), $MachinePrecision], N[(N[(0.125 * x), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.2e-80 or 1.09999999999999999e188 < 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. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. div-inv 100.0%

         \[\leadsto \left(\left(-\color{blue}{z \cdot \frac{1}{2}}\right) \cdot y + 0.125 \cdot x\right) + t \]

      6. distribute-rgt-neg-in 100.0%

         \[\leadsto \left(\color{blue}{\left(z \cdot \left(-\frac{1}{2}\right)\right)} \cdot y + 0.125 \cdot x\right) + t \]

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. add-sqr-sqrt 72.4%

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

      10. sqrt-unprod 57.1%

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

      11. swap-sqr 57.1%

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

      12. metadata-eval 57.1%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \color{blue}{0.25}} \cdot y + 0.125 \cdot x\right) + t \]

      13. metadata-eval 57.1%

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

      14. metadata-eval 57.1%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot 0.5\right)} \cdot y + 0.125 \cdot x\right) + t \]

      15. metadata-eval 57.1%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{2}}\right)} \cdot y + 0.125 \cdot x\right) + t \]

      16. swap-sqr 57.1%

          \[\leadsto \left(\sqrt{\color{blue}{\left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right)}} \cdot y + 0.125 \cdot x\right) + t \]

      17. div-inv 57.1%

          \[\leadsto \left(\sqrt{\color{blue}{\frac{z}{2}} \cdot \left(z \cdot \frac{1}{2}\right)} \cdot y + 0.125 \cdot x\right) + t \]

      18. div-inv 57.1%

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

      19. sqrt-unprod 7.4%

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

      20. add-sqr-sqrt 45.1%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in z around inf 53.9%

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

      1. associate-*r* 53.9%

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

      2. *-commutative 53.9%

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

   9. Simplified 53.9%

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

   if -7.2e-80 < z < 1.09999999999999999e188

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 78.7%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-80} \lor \neg \left(z \leq 1.1 \cdot 10^{+188}\right):\\ \;\;\;\;z \cdot \left(-0.5 \cdot y\right)\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x + t\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 49.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+75}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-53}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -4.4e+75) t (if (<= t 1.95e-53) (* 0.125 x) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -4.4e+75) {
		tmp = t;
	} else if (t <= 1.95e-53) {
		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 <= (-4.4d+75)) then
        tmp = t
    else if (t <= 1.95d-53) 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 <= -4.4e+75) {
		tmp = t;
	} else if (t <= 1.95e-53) {
		tmp = 0.125 * x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -4.4e+75:
		tmp = t
	elif t <= 1.95e-53:
		tmp = 0.125 * x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -4.4e+75)
		tmp = t;
	elseif (t <= 1.95e-53)
		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 <= -4.4e+75)
		tmp = t;
	elseif (t <= 1.95e-53)
		tmp = 0.125 * x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -4.4e+75], t, If[LessEqual[t, 1.95e-53], N[(0.125 * x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if t < -4.40000000000000024e75 or 1.9500000000000001e-53 < 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. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in t around inf 64.7%

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

   if -4.40000000000000024e75 < t < 1.9500000000000001e-53

   1. Initial program 100.0%

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

      1. associate-+l- 100.0%

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

      2. *-commutative 100.0%

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

      3. associate-+l- 100.0%

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

      4. metadata-eval 100.0%

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

      5. *-commutative 100.0%

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

      6. associate-/l* 100.0%

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

   3. Simplified 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. *-commutative 100.0%

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

      4. distribute-lft-neg-in 100.0%

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

      5. div-inv 100.0%

         \[\leadsto \left(\left(-\color{blue}{z \cdot \frac{1}{2}}\right) \cdot y + 0.125 \cdot x\right) + t \]

      6. distribute-rgt-neg-in 100.0%

         \[\leadsto \left(\color{blue}{\left(z \cdot \left(-\frac{1}{2}\right)\right)} \cdot y + 0.125 \cdot x\right) + t \]

      7. metadata-eval 100.0%

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

      8. metadata-eval 100.0%

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

      9. add-sqr-sqrt 49.5%

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

      10. sqrt-unprod 66.1%

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

      11. swap-sqr 66.1%

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

      12. metadata-eval 66.1%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \color{blue}{0.25}} \cdot y + 0.125 \cdot x\right) + t \]

      13. metadata-eval 66.1%

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

      14. metadata-eval 66.1%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \left(\color{blue}{\frac{1}{2}} \cdot 0.5\right)} \cdot y + 0.125 \cdot x\right) + t \]

      15. metadata-eval 66.1%

          \[\leadsto \left(\sqrt{\left(z \cdot z\right) \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{2}}\right)} \cdot y + 0.125 \cdot x\right) + t \]

      16. swap-sqr 66.1%

          \[\leadsto \left(\sqrt{\color{blue}{\left(z \cdot \frac{1}{2}\right) \cdot \left(z \cdot \frac{1}{2}\right)}} \cdot y + 0.125 \cdot x\right) + t \]

      17. div-inv 66.1%

          \[\leadsto \left(\sqrt{\color{blue}{\frac{z}{2}} \cdot \left(z \cdot \frac{1}{2}\right)} \cdot y + 0.125 \cdot x\right) + t \]

      18. div-inv 66.1%

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

      19. sqrt-unprod 25.0%

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

      20. add-sqr-sqrt 49.8%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around inf 43.6%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+75}:\\ \;\;\;\;t\\ \mathbf{elif}\;t \leq 1.95 \cdot 10^{-53}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. *-commutative 100.0%

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

   3. associate-+l- 100.0%

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

   4. metadata-eval 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-/l* 100.0%

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

3. Simplified 100.0%

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

5. Final simplification 100.0%

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

Alternative 8: 33.4% 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. associate-+l- 100.0%

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

   2. *-commutative 100.0%

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

   3. associate-+l- 100.0%

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

   4. metadata-eval 100.0%

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

   5. *-commutative 100.0%

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

   6. associate-/l* 100.0%

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

3. Simplified 100.0%

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

5. Taylor expanded in t around inf 38.0%

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

6. Final simplification 38.0%

   \[\leadsto t \]
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 2024044 
(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))