Numeric.AD.Rank1.Halley:findZero from ad-4.2.4

Percentage Accurate: 81.3% → 92.8%

Time: 9.4s

Alternatives: 6

Speedup: 1.4×

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 81.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 92.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y \cdot 2\right) \cdot z\\ t_2 := z \cdot \left(2 \cdot z\right)\\ \mathbf{if}\;\frac{t\_1}{t\_2 - y \cdot t} \leq 10^{+179}:\\ \;\;\;\;x + \frac{t\_1}{y \cdot t - t\_2}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (* y 2.0) z)) (t_2 (* z (* 2.0 z))))
   (if (<= (/ t_1 (- t_2 (* y t))) 1e+179)
     (+ x (/ t_1 (- (* y t) t_2)))
     (- x (/ y z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * 2.0) * z;
	double t_2 = z * (2.0 * z);
	double tmp;
	if ((t_1 / (t_2 - (y * t))) <= 1e+179) {
		tmp = x + (t_1 / ((y * t) - t_2));
	} else {
		tmp = x - (y / z);
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = (y * 2.0d0) * z
    t_2 = z * (2.0d0 * z)
    if ((t_1 / (t_2 - (y * t))) <= 1d+179) then
        tmp = x + (t_1 / ((y * t) - t_2))
    else
        tmp = x - (y / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * 2.0) * z;
	double t_2 = z * (2.0 * z);
	double tmp;
	if ((t_1 / (t_2 - (y * t))) <= 1e+179) {
		tmp = x + (t_1 / ((y * t) - t_2));
	} else {
		tmp = x - (y / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * 2.0) * z
	t_2 = z * (2.0 * z)
	tmp = 0
	if (t_1 / (t_2 - (y * t))) <= 1e+179:
		tmp = x + (t_1 / ((y * t) - t_2))
	else:
		tmp = x - (y / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * 2.0) * z)
	t_2 = Float64(z * Float64(2.0 * z))
	tmp = 0.0
	if (Float64(t_1 / Float64(t_2 - Float64(y * t))) <= 1e+179)
		tmp = Float64(x + Float64(t_1 / Float64(Float64(y * t) - t_2)));
	else
		tmp = Float64(x - Float64(y / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * 2.0) * z;
	t_2 = z * (2.0 * z);
	tmp = 0.0;
	if ((t_1 / (t_2 - (y * t))) <= 1e+179)
		tmp = x + (t_1 / ((y * t) - t_2));
	else
		tmp = x - (y / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * 2.0), $MachinePrecision] * z), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(2.0 * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(t$95$1 / N[(t$95$2 - N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1e+179], N[(x + N[(t$95$1 / N[(N[(y * t), $MachinePrecision] - t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t))) < 9.9999999999999998e178

   1. Initial program 94.0%

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

   if 9.9999999999999998e178 < (/.f64 (*.f64 (*.f64 y #s(literal 2 binary64)) z) (-.f64 (*.f64 (*.f64 z #s(literal 2 binary64)) z) (*.f64 y t)))

   1. Initial program 2.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{x - \frac{y}{z}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y}{z}} \]

      4. lower-/.f64 70.7

         \[\leadsto x - \color{blue}{\frac{y}{z}} \]

   5. Simplified 70.7%

      \[\leadsto \color{blue}{x - \frac{y}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y \cdot 2\right) \cdot z}{z \cdot \left(2 \cdot z\right) - y \cdot t} \leq 10^{+179}:\\ \;\;\;\;x + \frac{\left(y \cdot 2\right) \cdot z}{y \cdot t - z \cdot \left(2 \cdot z\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 89.2% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 950:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, 2, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.2e+32) t_1 (if (<= z 950.0) (fma (/ z t) 2.0 x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.2e+32) {
		tmp = t_1;
	} else if (z <= 950.0) {
		tmp = fma((z / t), 2.0, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -1.2e+32)
		tmp = t_1;
	elseif (z <= 950.0)
		tmp = fma(Float64(z / t), 2.0, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.2e+32], t$95$1, If[LessEqual[z, 950.0], N[(N[(z / t), $MachinePrecision] * 2.0 + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.19999999999999996e32 or 950 < z

   1. Initial program 68.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{x - \frac{y}{z}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y}{z}} \]

      4. lower-/.f64 86.5

         \[\leadsto x - \color{blue}{\frac{y}{z}} \]

   5. Simplified 86.5%

      \[\leadsto \color{blue}{x - \frac{y}{z}} \]

   if -1.19999999999999996e32 < z < 950

   1. Initial program 90.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto x - \color{blue}{-2 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto x - \color{blue}{\frac{-2 \cdot z}{t}} \]

      2. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{-2 \cdot z}{t}} \]

      3. lower-*.f64 92.9

         \[\leadsto x - \frac{\color{blue}{-2 \cdot z}}{t} \]

   5. Simplified 92.9%

      \[\leadsto x - \color{blue}{\frac{-2 \cdot z}{t}} \]
   6. Step-by-step derivation

      1. lift-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{-2 \cdot z}}{t} \]

      2. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{-2 \cdot z}{t}} \]

      3. sub-neg N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r/ N/A

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

      9. metadata-eval N/A

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

      10. frac-2neg N/A

          \[\leadsto x + z \cdot \color{blue}{\frac{2}{t}} \]

      11. lift-/.f64 N/A

          \[\leadsto x + z \cdot \color{blue}{\frac{2}{t}} \]

      12. +-commutative N/A

          \[\leadsto \color{blue}{z \cdot \frac{2}{t} + x} \]

      13. lift-/.f64 N/A

          \[\leadsto z \cdot \color{blue}{\frac{2}{t}} + x \]

      14. clear-num N/A

          \[\leadsto z \cdot \color{blue}{\frac{1}{\frac{t}{2}}} + x \]

      15. associate-*r/ N/A

          \[\leadsto \color{blue}{\frac{z \cdot 1}{\frac{t}{2}}} + x \]

      16. div-inv N/A

          \[\leadsto \frac{z \cdot 1}{\color{blue}{t \cdot \frac{1}{2}}} + x \]

      17. times-frac N/A

          \[\leadsto \color{blue}{\frac{z}{t} \cdot \frac{1}{\frac{1}{2}}} + x \]

      18. metadata-eval N/A

          \[\leadsto \frac{z}{t} \cdot \frac{1}{\color{blue}{\frac{1}{2}}} + x \]

      19. metadata-eval N/A

          \[\leadsto \frac{z}{t} \cdot \color{blue}{2} + x \]

      20. lower-fma.f64 N/A

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

      21. lower-/.f64 92.9

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

   7. Applied egg-rr 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{z}{t}, 2, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 89.1% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -1.2 \cdot 10^{+32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 950:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{2}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z))))
   (if (<= z -1.2e+32) t_1 (if (<= z 950.0) (fma z (/ 2.0 t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double tmp;
	if (z <= -1.2e+32) {
		tmp = t_1;
	} else if (z <= 950.0) {
		tmp = fma(z, (2.0 / t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -1.2e+32)
		tmp = t_1;
	elseif (z <= 950.0)
		tmp = fma(z, Float64(2.0 / t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.2e+32], t$95$1, If[LessEqual[z, 950.0], N[(z * N[(2.0 / t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.19999999999999996e32 or 950 < z

   1. Initial program 68.2%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{x - \frac{y}{z}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y}{z}} \]

      4. lower-/.f64 86.5

         \[\leadsto x - \color{blue}{\frac{y}{z}} \]

   5. Simplified 86.5%

      \[\leadsto \color{blue}{x - \frac{y}{z}} \]

   if -1.19999999999999996e32 < z < 950

   1. Initial program 90.6%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{x - -2 \cdot \frac{z}{t}} \]
   4. Step-by-step derivation

      1. cancel-sign-sub-inv N/A

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

      2. metadata-eval N/A

         \[\leadsto x + \color{blue}{2} \cdot \frac{z}{t} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{2 \cdot \frac{z}{t} + x} \]

      4. associate-*r/ N/A

         \[\leadsto \color{blue}{\frac{2 \cdot z}{t}} + x \]

      5. *-commutative N/A

         \[\leadsto \frac{\color{blue}{z \cdot 2}}{t} + x \]

      6. associate-*r/ N/A

         \[\leadsto \color{blue}{z \cdot \frac{2}{t}} + x \]

      7. metadata-eval N/A

         \[\leadsto z \cdot \frac{\color{blue}{2 \cdot 1}}{t} + x \]

      8. associate-*r/ N/A

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

      9. lower-fma.f64 N/A

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

      10. associate-*r/ N/A

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

      11. metadata-eval N/A

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

      12. lower-/.f64 92.9

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

   5. Simplified 92.9%

      \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{2}{t}, x\right)} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 52.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y}{z}\\ \mathbf{if}\;x \leq -3.8 \cdot 10^{-137}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ y z))))
   (if (<= x -3.8e-137) t_1 (if (<= x 7.5e-111) (/ y (- z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (y / z);
	double tmp;
	if (x <= -3.8e-137) {
		tmp = t_1;
	} else if (x <= 7.5e-111) {
		tmp = y / -z;
	} 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 = x + (y / z)
    if (x <= (-3.8d-137)) then
        tmp = t_1
    else if (x <= 7.5d-111) then
        tmp = y / -z
    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 = x + (y / z);
	double tmp;
	if (x <= -3.8e-137) {
		tmp = t_1;
	} else if (x <= 7.5e-111) {
		tmp = y / -z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (y / z)
	tmp = 0
	if x <= -3.8e-137:
		tmp = t_1
	elif x <= 7.5e-111:
		tmp = y / -z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(y / z))
	tmp = 0.0
	if (x <= -3.8e-137)
		tmp = t_1;
	elseif (x <= 7.5e-111)
		tmp = Float64(y / Float64(-z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x + (y / z);
	tmp = 0.0;
	if (x <= -3.8e-137)
		tmp = t_1;
	elseif (x <= 7.5e-111)
		tmp = y / -z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -3.8e-137], t$95$1, If[LessEqual[x, 7.5e-111], N[(y / (-z)), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.79999999999999999e-137 or 7.49999999999999965e-111 < x

   1. Initial program 83.3%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{x - \frac{y}{z}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y}{z}} \]

      4. lower-/.f64 66.8

         \[\leadsto x - \color{blue}{\frac{y}{z}} \]

   5. Simplified 66.8%

      \[\leadsto \color{blue}{x - \frac{y}{z}} \]
   6. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{y}{z}} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{x + \left(\mathsf{neg}\left(\frac{y}{z}\right)\right)} \]

      3. +-commutative N/A

         \[\leadsto \color{blue}{\left(\mathsf{neg}\left(\frac{y}{z}\right)\right) + x} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\mathsf{neg}\left(\color{blue}{\frac{y}{z}}\right)\right) + x \]

      5. div-inv N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. lower-neg.f64 N/A

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

      9. lower-/.f64 66.7

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

   7. Applied egg-rr 66.7%

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

      1. inv-pow N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), \color{blue}{{z}^{-1}}, x\right) \]

      2. pow-to-exp N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), \color{blue}{e^{\log z \cdot -1}}, x\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), e^{\color{blue}{-1 \cdot \log z}}, x\right) \]

      4. log-pow N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), e^{\color{blue}{\log \left({z}^{-1}\right)}}, x\right) \]

      5. inv-pow N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), e^{\log \color{blue}{\left(\frac{1}{z}\right)}}, x\right) \]

      6. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), e^{\log \color{blue}{\left(\frac{1}{z}\right)}}, x\right) \]

      7. lower-exp.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), \color{blue}{e^{\log \left(\frac{1}{z}\right)}}, x\right) \]

      8. lift-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), e^{\log \color{blue}{\left(\frac{1}{z}\right)}}, x\right) \]

      9. log-rec N/A

         \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), e^{\color{blue}{\mathsf{neg}\left(\log z\right)}}, x\right) \]

      10. lower-neg.f64 N/A

          \[\leadsto \mathsf{fma}\left(\mathsf{neg}\left(y\right), e^{\color{blue}{\mathsf{neg}\left(\log z\right)}}, x\right) \]

      11. lower-log.f64 35.7

          \[\leadsto \mathsf{fma}\left(-y, e^{-\color{blue}{\log z}}, x\right) \]

   9. Applied egg-rr 35.7%

      \[\leadsto \mathsf{fma}\left(-y, \color{blue}{e^{-\log z}}, x\right) \]

   11. Applied egg-rr 61.8%

       \[\leadsto \color{blue}{\frac{y}{z} + x} \]

   if -3.79999999999999999e-137 < x < 7.49999999999999965e-111

   1. Initial program 70.5%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{z}} \]
   4. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{z}\right)\right)} \]

      2. sub-neg N/A

         \[\leadsto \color{blue}{x - \frac{y}{z}} \]

      3. lower--.f64 N/A

         \[\leadsto \color{blue}{x - \frac{y}{z}} \]

      4. lower-/.f64 49.7

         \[\leadsto x - \color{blue}{\frac{y}{z}} \]

   5. Simplified 49.7%

      \[\leadsto \color{blue}{x - \frac{y}{z}} \]

   6. Taylor expanded in x around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{y}{z}} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

         \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y}{z}\right)} \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \color{blue}{\frac{y}{\mathsf{neg}\left(z\right)}} \]

      3. mul-1-neg N/A

         \[\leadsto \frac{y}{\color{blue}{-1 \cdot z}} \]

      4. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{y}{-1 \cdot z}} \]

      5. mul-1-neg N/A

         \[\leadsto \frac{y}{\color{blue}{\mathsf{neg}\left(z\right)}} \]

      6. lower-neg.f64 32.9

         \[\leadsto \frac{y}{\color{blue}{-z}} \]

   8. Simplified 32.9%

      \[\leadsto \color{blue}{\frac{y}{-z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.8 \cdot 10^{-137}:\\ \;\;\;\;x + \frac{y}{z}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-111}:\\ \;\;\;\;\frac{y}{-z}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 62.4% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{z} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (y / z)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{z}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{z}\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{x - \frac{y}{z}} \]

   3. lower--.f64 N/A

      \[\leadsto \color{blue}{x - \frac{y}{z}} \]

   4. lower-/.f64 61.6

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]

5. Simplified 61.6%

   \[\leadsto \color{blue}{x - \frac{y}{z}} \]
6. Add Preprocessing

Alternative 6: 14.8% accurate, 3.1× speedup

Math

\[\begin{array}{l} \\ \frac{y}{-z} \end{array} \]

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = y / -z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(y / (-z)), $MachinePrecision]

Derivation

1. Initial program 79.4%

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

3. Taylor expanded in y around 0

   \[\leadsto \color{blue}{x + -1 \cdot \frac{y}{z}} \]
4. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(\frac{y}{z}\right)\right)} \]

   2. sub-neg N/A

      \[\leadsto \color{blue}{x - \frac{y}{z}} \]

   3. lower--.f64 N/A

      \[\leadsto \color{blue}{x - \frac{y}{z}} \]

   4. lower-/.f64 61.6

      \[\leadsto x - \color{blue}{\frac{y}{z}} \]

5. Simplified 61.6%

   \[\leadsto \color{blue}{x - \frac{y}{z}} \]

6. Taylor expanded in x around 0

   \[\leadsto \color{blue}{-1 \cdot \frac{y}{z}} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto \color{blue}{\mathsf{neg}\left(\frac{y}{z}\right)} \]

   2. distribute-neg-frac2 N/A

      \[\leadsto \color{blue}{\frac{y}{\mathsf{neg}\left(z\right)}} \]

   3. mul-1-neg N/A

      \[\leadsto \frac{y}{\color{blue}{-1 \cdot z}} \]

   4. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{y}{-1 \cdot z}} \]

   5. mul-1-neg N/A

      \[\leadsto \frac{y}{\color{blue}{\mathsf{neg}\left(z\right)}} \]

   6. lower-neg.f64 15.0

      \[\leadsto \frac{y}{\color{blue}{-z}} \]

8. Simplified 15.0%

   \[\leadsto \color{blue}{\frac{y}{-z}} \]
9. Add Preprocessing

Developer Target 1: 99.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ x - \frac{1}{\frac{z}{y} - \frac{\frac{t}{2}}{z}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z)))))

C

double code(double x, double y, double z, double t) {
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (1.0d0 / ((z / y) - ((t / 2.0d0) / z)))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)));
}

Python

def code(x, y, z, t):
	return x - (1.0 / ((z / y) - ((t / 2.0) / z)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(1.0 / Float64(Float64(z / y) - Float64(Float64(t / 2.0) / z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (1.0 / ((z / y) - ((t / 2.0) / z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(1.0 / N[(N[(z / y), $MachinePrecision] - N[(N[(t / 2.0), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Reproduce

herbie shell --seed 2024208 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (/ 1 (- (/ z y) (/ (/ t 2) z)))))

  (- x (/ (* (* y 2.0) z) (- (* (* z 2.0) z) (* y t)))))