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

Percentage Accurate: 100.0% → 100.0%

Time: 5.2s

Alternatives: 7

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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. remove-double-neg 100.0%

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

   2. sub-neg 100.0%

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

   3. sub-neg 100.0%

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

   4. +-commutative 100.0%

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

   5. associate--l+ 100.0%

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

   6. *-commutative 100.0%

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

   7. associate-*r/ 100.0%

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

   8. distribute-rgt-neg-in 100.0%

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

   9. fma-def 100.0%

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

   10. neg-mul-1 100.0%

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

   11. *-commutative 100.0%

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

   12. associate-*l/ 100.0%

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

   13. associate-/l* 100.0%

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

   14. metadata-eval 100.0%

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

   15. fma-neg 100.0%

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

   16. remove-double-neg 100.0%

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

   17. metadata-eval 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

   \[\leadsto \mathsf{fma}\left(z, \frac{y}{-2}, \mathsf{fma}\left(0.125, x, t\right)\right) \]

Alternative 2: 48.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(y \cdot -0.5\right)\\ \mathbf{if}\;z \leq -2.75 \cdot 10^{-122}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-228}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-198}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-57}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-34}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;z \leq 35000000000:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (* y -0.5))))
   (if (<= z -2.75e-122)
     t_1
     (if (<= z 7.5e-228)
       t
       (if (<= z 3.8e-198)
         (* 0.125 x)
         (if (<= z 3.5e-57)
           t
           (if (<= z 5e-34) (* 0.125 x) (if (<= z 35000000000.0) t t_1))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (y * -0.5);
	double tmp;
	if (z <= -2.75e-122) {
		tmp = t_1;
	} else if (z <= 7.5e-228) {
		tmp = t;
	} else if (z <= 3.8e-198) {
		tmp = 0.125 * x;
	} else if (z <= 3.5e-57) {
		tmp = t;
	} else if (z <= 5e-34) {
		tmp = 0.125 * x;
	} else if (z <= 35000000000.0) {
		tmp = t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (y * (-0.5d0))
    if (z <= (-2.75d-122)) then
        tmp = t_1
    else if (z <= 7.5d-228) then
        tmp = t
    else if (z <= 3.8d-198) then
        tmp = 0.125d0 * x
    else if (z <= 3.5d-57) then
        tmp = t
    else if (z <= 5d-34) then
        tmp = 0.125d0 * x
    else if (z <= 35000000000.0d0) then
        tmp = t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (y * -0.5);
	double tmp;
	if (z <= -2.75e-122) {
		tmp = t_1;
	} else if (z <= 7.5e-228) {
		tmp = t;
	} else if (z <= 3.8e-198) {
		tmp = 0.125 * x;
	} else if (z <= 3.5e-57) {
		tmp = t;
	} else if (z <= 5e-34) {
		tmp = 0.125 * x;
	} else if (z <= 35000000000.0) {
		tmp = t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (y * -0.5)
	tmp = 0
	if z <= -2.75e-122:
		tmp = t_1
	elif z <= 7.5e-228:
		tmp = t
	elif z <= 3.8e-198:
		tmp = 0.125 * x
	elif z <= 3.5e-57:
		tmp = t
	elif z <= 5e-34:
		tmp = 0.125 * x
	elif z <= 35000000000.0:
		tmp = t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(y * -0.5))
	tmp = 0.0
	if (z <= -2.75e-122)
		tmp = t_1;
	elseif (z <= 7.5e-228)
		tmp = t;
	elseif (z <= 3.8e-198)
		tmp = Float64(0.125 * x);
	elseif (z <= 3.5e-57)
		tmp = t;
	elseif (z <= 5e-34)
		tmp = Float64(0.125 * x);
	elseif (z <= 35000000000.0)
		tmp = t;
	else
		tmp = 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 <= -2.75e-122)
		tmp = t_1;
	elseif (z <= 7.5e-228)
		tmp = t;
	elseif (z <= 3.8e-198)
		tmp = 0.125 * x;
	elseif (z <= 3.5e-57)
		tmp = t;
	elseif (z <= 5e-34)
		tmp = 0.125 * x;
	elseif (z <= 35000000000.0)
		tmp = t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.75e-122], t$95$1, If[LessEqual[z, 7.5e-228], t, If[LessEqual[z, 3.8e-198], N[(0.125 * x), $MachinePrecision], If[LessEqual[z, 3.5e-57], t, If[LessEqual[z, 5e-34], N[(0.125 * x), $MachinePrecision], If[LessEqual[z, 35000000000.0], t, t$95$1]]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.75000000000000026e-122 or 3.5e10 < 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 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. remove-double-neg 100.0%

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

      13. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 55.0%

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

      1. associate-*r* 55.0%

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

      2. *-commutative 55.0%

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

   6. Simplified 55.0%

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

   if -2.75000000000000026e-122 < z < 7.4999999999999999e-228 or 3.8000000000000002e-198 < z < 3.49999999999999991e-57 or 5.0000000000000003e-34 < z < 3.5e10

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. remove-double-neg 100.0%

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

      13. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 55.8%

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

   if 7.4999999999999999e-228 < z < 3.8000000000000002e-198 or 3.49999999999999991e-57 < z < 5.0000000000000003e-34

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. remove-double-neg 100.0%

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

      13. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 77.5%

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

4. Final simplification 56.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.75 \cdot 10^{-122}:\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{-228}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{-198}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{-57}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 5 \cdot 10^{-34}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;z \leq 35000000000:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \end{array} \]

Alternative 3: 88.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+38} \lor \neg \left(z \cdot y \leq 10^{+83}\right):\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z y) -1e+38) (not (<= (* z y) 1e+83)))
   (- t (* (* z y) 0.5))
   (+ t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -1e+38) || !((z * y) <= 1e+83)) {
		tmp = t - ((z * y) * 0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * y) <= (-1d+38)) .or. (.not. ((z * y) <= 1d+83))) then
        tmp = t - ((z * y) * 0.5d0)
    else
        tmp = t + (0.125d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * y) <= -1e+38) || !((z * y) <= 1e+83)) {
		tmp = t - ((z * y) * 0.5);
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((z * y) <= -1e+38) or not ((z * y) <= 1e+83):
		tmp = t - ((z * y) * 0.5)
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * y) <= -1e+38) || !(Float64(z * y) <= 1e+83))
		tmp = Float64(t - Float64(Float64(z * y) * 0.5));
	else
		tmp = Float64(t + Float64(0.125 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * y) <= -1e+38) || ~(((z * y) <= 1e+83)))
		tmp = t - ((z * y) * 0.5);
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * y), $MachinePrecision], -1e+38], N[Not[LessEqual[N[(z * y), $MachinePrecision], 1e+83]], $MachinePrecision]], N[(t - N[(N[(z * y), $MachinePrecision] * 0.5), $MachinePrecision]), $MachinePrecision], N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y z) < -9.99999999999999977e37 or 1.00000000000000003e83 < (*.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 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. remove-double-neg 100.0%

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

      13. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around 0 91.6%

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

   if -9.99999999999999977e37 < (*.f64 y z) < 1.00000000000000003e83

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. remove-double-neg 100.0%

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

      13. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 95.8%

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

4. Final simplification 93.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot y \leq -1 \cdot 10^{+38} \lor \neg \left(z \cdot y \leq 10^{+83}\right):\\ \;\;\;\;t - \left(z \cdot y\right) \cdot 0.5\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]

Alternative 4: 72.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(y \cdot -0.5\right)\\ \mathbf{if}\;y \leq -1.35 \cdot 10^{+227}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{+209}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+112} \lor \neg \left(y \leq 7.8 \cdot 10^{+15}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (* y -0.5))))
   (if (<= y -1.35e+227)
     t_1
     (if (<= y -4.7e+209)
       t
       (if (or (<= y -3.7e+112) (not (<= y 7.8e+15)))
         t_1
         (+ t (* 0.125 x)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (y * -0.5);
	double tmp;
	if (y <= -1.35e+227) {
		tmp = t_1;
	} else if (y <= -4.7e+209) {
		tmp = t;
	} else if ((y <= -3.7e+112) || !(y <= 7.8e+15)) {
		tmp = t_1;
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (y * (-0.5d0))
    if (y <= (-1.35d+227)) then
        tmp = t_1
    else if (y <= (-4.7d+209)) then
        tmp = t
    else if ((y <= (-3.7d+112)) .or. (.not. (y <= 7.8d+15))) then
        tmp = t_1
    else
        tmp = t + (0.125d0 * x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = z * (y * -0.5);
	double tmp;
	if (y <= -1.35e+227) {
		tmp = t_1;
	} else if (y <= -4.7e+209) {
		tmp = t;
	} else if ((y <= -3.7e+112) || !(y <= 7.8e+15)) {
		tmp = t_1;
	} else {
		tmp = t + (0.125 * x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = z * (y * -0.5)
	tmp = 0
	if y <= -1.35e+227:
		tmp = t_1
	elif y <= -4.7e+209:
		tmp = t
	elif (y <= -3.7e+112) or not (y <= 7.8e+15):
		tmp = t_1
	else:
		tmp = t + (0.125 * x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(y * -0.5))
	tmp = 0.0
	if (y <= -1.35e+227)
		tmp = t_1;
	elseif (y <= -4.7e+209)
		tmp = t;
	elseif ((y <= -3.7e+112) || !(y <= 7.8e+15))
		tmp = t_1;
	else
		tmp = Float64(t + Float64(0.125 * x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = z * (y * -0.5);
	tmp = 0.0;
	if (y <= -1.35e+227)
		tmp = t_1;
	elseif (y <= -4.7e+209)
		tmp = t;
	elseif ((y <= -3.7e+112) || ~((y <= 7.8e+15)))
		tmp = t_1;
	else
		tmp = t + (0.125 * x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(y * -0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -1.35e+227], t$95$1, If[LessEqual[y, -4.7e+209], t, If[Or[LessEqual[y, -3.7e+112], N[Not[LessEqual[y, 7.8e+15]], $MachinePrecision]], t$95$1, N[(t + N[(0.125 * x), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.3499999999999999e227 or -4.7000000000000001e209 < y < -3.70000000000000004e112 or 7.8e15 < y

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. remove-double-neg 100.0%

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

      13. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around inf 71.4%

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

      1. associate-*r* 71.4%

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

      2. *-commutative 71.4%

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

   6. Simplified 71.4%

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

   if -1.3499999999999999e227 < y < -4.7000000000000001e209

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. remove-double-neg 100.0%

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

      13. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 100.0%

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

   if -3.70000000000000004e112 < y < 7.8e15

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. remove-double-neg 100.0%

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

      13. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 84.5%

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

4. Final simplification 79.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.35 \cdot 10^{+227}:\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{+209}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{+112} \lor \neg \left(y \leq 7.8 \cdot 10^{+15}\right):\\ \;\;\;\;z \cdot \left(y \cdot -0.5\right)\\ \mathbf{else}:\\ \;\;\;\;t + 0.125 \cdot x\\ \end{array} \]

Alternative 5: 100.0% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = t + ((0.125d0 * x) - (z * (y / 2.0d0)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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

   2. +-commutative 100.0%

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

   3. neg-sub0 100.0%

      \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

   4. associate-+l- 100.0%

      \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

   5. sub-neg 100.0%

      \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

   6. +-commutative 100.0%

      \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

   7. associate--r+ 100.0%

      \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

   8. neg-sub0 100.0%

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

   9. distribute-rgt-neg-in 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. metadata-eval 100.0%

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

   12. remove-double-neg 100.0%

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

   13. associate-*l/ 100.0%

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

3. Simplified 100.0%

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

4. Final simplification 100.0%

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

Alternative 6: 49.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+120}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+50}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.4e+120) (* 0.125 x) (if (<= x 3.8e+50) t (* 0.125 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.4e+120) {
		tmp = 0.125 * x;
	} else if (x <= 3.8e+50) {
		tmp = t;
	} else {
		tmp = 0.125 * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-1.4d+120)) then
        tmp = 0.125d0 * x
    else if (x <= 3.8d+50) then
        tmp = t
    else
        tmp = 0.125d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.4e+120) {
		tmp = 0.125 * x;
	} else if (x <= 3.8e+50) {
		tmp = t;
	} else {
		tmp = 0.125 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.4e+120:
		tmp = 0.125 * x
	elif x <= 3.8e+50:
		tmp = t
	else:
		tmp = 0.125 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.4e+120)
		tmp = Float64(0.125 * x);
	elseif (x <= 3.8e+50)
		tmp = t;
	else
		tmp = Float64(0.125 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.4e+120)
		tmp = 0.125 * x;
	elseif (x <= 3.8e+50)
		tmp = t;
	else
		tmp = 0.125 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.4e+120], N[(0.125 * x), $MachinePrecision], If[LessEqual[x, 3.8e+50], t, N[(0.125 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.4e120 or 3.79999999999999987e50 < x

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. remove-double-neg 100.0%

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

      13. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in x around inf 66.8%

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

   if -1.4e120 < x < 3.79999999999999987e50

   1. Initial program 100.0%

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

      1. sub-neg 100.0%

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

      2. +-commutative 100.0%

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

      3. neg-sub0 100.0%

         \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

      4. associate-+l- 100.0%

         \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

      5. sub-neg 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

      6. +-commutative 100.0%

         \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

      7. associate--r+ 100.0%

         \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

      8. neg-sub0 100.0%

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

      9. distribute-rgt-neg-in 100.0%

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

      10. distribute-rgt-neg-in 100.0%

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

      11. metadata-eval 100.0%

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

      12. remove-double-neg 100.0%

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

      13. associate-*l/ 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around inf 45.9%

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

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.4 \cdot 10^{+120}:\\ \;\;\;\;0.125 \cdot x\\ \mathbf{elif}\;x \leq 3.8 \cdot 10^{+50}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;0.125 \cdot x\\ \end{array} \]

Alternative 7: 33.2% 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 100.0%

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

   2. +-commutative 100.0%

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

   3. neg-sub0 100.0%

      \[\leadsto \left(\color{blue}{\left(0 - \frac{y \cdot z}{2}\right)} + \frac{1}{8} \cdot x\right) + t \]

   4. associate-+l- 100.0%

      \[\leadsto \color{blue}{\left(0 - \left(\frac{y \cdot z}{2} - \frac{1}{8} \cdot x\right)\right)} + t \]

   5. sub-neg 100.0%

      \[\leadsto \left(0 - \color{blue}{\left(\frac{y \cdot z}{2} + \left(-\frac{1}{8} \cdot x\right)\right)}\right) + t \]

   6. +-commutative 100.0%

      \[\leadsto \left(0 - \color{blue}{\left(\left(-\frac{1}{8} \cdot x\right) + \frac{y \cdot z}{2}\right)}\right) + t \]

   7. associate--r+ 100.0%

      \[\leadsto \color{blue}{\left(\left(0 - \left(-\frac{1}{8} \cdot x\right)\right) - \frac{y \cdot z}{2}\right)} + t \]

   8. neg-sub0 100.0%

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

   9. distribute-rgt-neg-in 100.0%

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

   10. distribute-rgt-neg-in 100.0%

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

   11. metadata-eval 100.0%

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

   12. remove-double-neg 100.0%

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

   13. associate-*l/ 100.0%

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

3. Simplified 100.0%

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

4. Taylor expanded in t around inf 34.2%

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

5. Final simplification 34.2%

   \[\leadsto t \]

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 2023252 
(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))