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

Percentage Accurate: 81.4% → 98.6%

Time: 7.8s

Alternatives: 8

Speedup: 1.9×

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: 81.4% 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: 98.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(y \cdot \frac{1}{\mathsf{fma}\left(-2, z, \frac{y}{\frac{z}{t}}\right)}, 2, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

3. Simplified 93.7%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y}{z \cdot -2 + \frac{y \cdot t}{z}}, 2, x\right)} \]
4. Step-by-step derivation

   1. div-inv 93.6%

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

   2. *-commutative 93.6%

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

   3. fma-def 93.6%

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

   4. associate-/l* 97.3%

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

5. Applied egg-rr 97.3%

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

6. Final simplification 97.3%

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

Alternative 2: 95.9% accurate, 0.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

3. Simplified 93.7%

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

4. Final simplification 93.7%

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

Alternative 3: 97.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{z}\\ t_2 := x - \frac{y \cdot 2}{\frac{z \cdot \left(z \cdot 2\right) - y \cdot t}{z}}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-143}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-118}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+144}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ y z)))
        (t_2 (- x (/ (* y 2.0) (/ (- (* z (* z 2.0)) (* y t)) z)))))
   (if (<= z -5.5e+154)
     t_1
     (if (<= z -2.6e-143)
       t_2
       (if (<= z 1.35e-118)
         (- x (* -2.0 (/ z t)))
         (if (<= z 2.1e+144) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (y / z);
	double t_2 = x - ((y * 2.0) / (((z * (z * 2.0)) - (y * t)) / z));
	double tmp;
	if (z <= -5.5e+154) {
		tmp = t_1;
	} else if (z <= -2.6e-143) {
		tmp = t_2;
	} else if (z <= 1.35e-118) {
		tmp = x - (-2.0 * (z / t));
	} else if (z <= 2.1e+144) {
		tmp = t_2;
	} 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) :: t_2
    real(8) :: tmp
    t_1 = x - (y / z)
    t_2 = x - ((y * 2.0d0) / (((z * (z * 2.0d0)) - (y * t)) / z))
    if (z <= (-5.5d+154)) then
        tmp = t_1
    else if (z <= (-2.6d-143)) then
        tmp = t_2
    else if (z <= 1.35d-118) then
        tmp = x - ((-2.0d0) * (z / t))
    else if (z <= 2.1d+144) then
        tmp = t_2
    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 t_2 = x - ((y * 2.0) / (((z * (z * 2.0)) - (y * t)) / z));
	double tmp;
	if (z <= -5.5e+154) {
		tmp = t_1;
	} else if (z <= -2.6e-143) {
		tmp = t_2;
	} else if (z <= 1.35e-118) {
		tmp = x - (-2.0 * (z / t));
	} else if (z <= 2.1e+144) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (y / z)
	t_2 = x - ((y * 2.0) / (((z * (z * 2.0)) - (y * t)) / z))
	tmp = 0
	if z <= -5.5e+154:
		tmp = t_1
	elif z <= -2.6e-143:
		tmp = t_2
	elif z <= 1.35e-118:
		tmp = x - (-2.0 * (z / t))
	elif z <= 2.1e+144:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(y / z))
	t_2 = Float64(x - Float64(Float64(y * 2.0) / Float64(Float64(Float64(z * Float64(z * 2.0)) - Float64(y * t)) / z)))
	tmp = 0.0
	if (z <= -5.5e+154)
		tmp = t_1;
	elseif (z <= -2.6e-143)
		tmp = t_2;
	elseif (z <= 1.35e-118)
		tmp = Float64(x - Float64(-2.0 * Float64(z / t)));
	elseif (z <= 2.1e+144)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x - (y / z);
	t_2 = x - ((y * 2.0) / (((z * (z * 2.0)) - (y * t)) / z));
	tmp = 0.0;
	if (z <= -5.5e+154)
		tmp = t_1;
	elseif (z <= -2.6e-143)
		tmp = t_2;
	elseif (z <= 1.35e-118)
		tmp = x - (-2.0 * (z / t));
	elseif (z <= 2.1e+144)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(x - N[(N[(y * 2.0), $MachinePrecision] / N[(N[(N[(z * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e+154], t$95$1, If[LessEqual[z, -2.6e-143], t$95$2, If[LessEqual[z, 1.35e-118], N[(x - N[(-2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.1e+144], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.5000000000000006e154 or 2.09999999999999996e144 < z

   1. Initial program 50.7%

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

      1. sub-neg 50.7%

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

      2. associate-/l* 69.6%

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

      3. distribute-neg-frac 69.6%

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

      4. distribute-lft-neg-out 69.6%

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

      5. associate-/r/ 69.6%

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

      6. distribute-lft-neg-out 69.6%

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

      7. distribute-rgt-neg-in 69.6%

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

      8. metadata-eval 69.6%

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

      9. *-commutative 69.6%

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

      10. associate-*l* 69.6%

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

      11. fma-neg 69.6%

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

   3. Simplified 69.6%

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

   4. Taylor expanded in y around 0 98.7%

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

      1. mul-1-neg 98.7%

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

      2. sub-neg 98.7%

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

   6. Simplified 98.7%

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

   if -5.5000000000000006e154 < z < -2.59999999999999987e-143 or 1.34999999999999997e-118 < z < 2.09999999999999996e144

   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. Step-by-step derivation

      1. remove-double-neg 90.6%

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

      2. neg-mul-1 90.6%

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

      3. *-commutative 90.6%

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

      4. *-commutative 90.6%

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

      5. neg-mul-1 90.6%

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

      6. remove-double-neg 90.6%

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

      7. associate-/l* 94.9%

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

      8. associate-*l* 94.9%

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

   3. Simplified 94.9%

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

   if -2.59999999999999987e-143 < z < 1.34999999999999997e-118

   1. Initial program 94.3%

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

      1. remove-double-neg 94.3%

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

      2. neg-mul-1 94.3%

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

      3. *-commutative 94.3%

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

      4. *-commutative 94.3%

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

      5. neg-mul-1 94.3%

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

      6. remove-double-neg 94.3%

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

      7. associate-/l* 90.4%

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

      8. associate-*l* 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in y around inf 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+154}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-143}:\\ \;\;\;\;x - \frac{y \cdot 2}{\frac{z \cdot \left(z \cdot 2\right) - y \cdot t}{z}}\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-118}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+144}:\\ \;\;\;\;x - \frac{y \cdot 2}{\frac{z \cdot \left(z \cdot 2\right) - y \cdot t}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]

Alternative 4: 96.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(z \cdot 2\right) - y \cdot t\\ t_2 := x - \frac{y}{z}\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+154}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-145}:\\ \;\;\;\;x - \frac{y \cdot 2}{\frac{t_1}{z}}\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-107}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+101}:\\ \;\;\;\;x - \frac{z \cdot \left(y \cdot 2\right)}{t_1}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z (* z 2.0)) (* y t))) (t_2 (- x (/ y z))))
   (if (<= z -5.5e+154)
     t_2
     (if (<= z -2e-145)
       (- x (/ (* y 2.0) (/ t_1 z)))
       (if (<= z 1.16e-107)
         (- x (* -2.0 (/ z t)))
         (if (<= z 4.8e+101) (- x (/ (* z (* y 2.0)) t_1)) t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * (z * 2.0)) - (y * t);
	double t_2 = x - (y / z);
	double tmp;
	if (z <= -5.5e+154) {
		tmp = t_2;
	} else if (z <= -2e-145) {
		tmp = x - ((y * 2.0) / (t_1 / z));
	} else if (z <= 1.16e-107) {
		tmp = x - (-2.0 * (z / t));
	} else if (z <= 4.8e+101) {
		tmp = x - ((z * (y * 2.0)) / t_1);
	} else {
		tmp = t_2;
	}
	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 = (z * (z * 2.0d0)) - (y * t)
    t_2 = x - (y / z)
    if (z <= (-5.5d+154)) then
        tmp = t_2
    else if (z <= (-2d-145)) then
        tmp = x - ((y * 2.0d0) / (t_1 / z))
    else if (z <= 1.16d-107) then
        tmp = x - ((-2.0d0) * (z / t))
    else if (z <= 4.8d+101) then
        tmp = x - ((z * (y * 2.0d0)) / t_1)
    else
        tmp = t_2
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * (z * 2.0)) - (y * t);
	double t_2 = x - (y / z);
	double tmp;
	if (z <= -5.5e+154) {
		tmp = t_2;
	} else if (z <= -2e-145) {
		tmp = x - ((y * 2.0) / (t_1 / z));
	} else if (z <= 1.16e-107) {
		tmp = x - (-2.0 * (z / t));
	} else if (z <= 4.8e+101) {
		tmp = x - ((z * (y * 2.0)) / t_1);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * (z * 2.0)) - (y * t)
	t_2 = x - (y / z)
	tmp = 0
	if z <= -5.5e+154:
		tmp = t_2
	elif z <= -2e-145:
		tmp = x - ((y * 2.0) / (t_1 / z))
	elif z <= 1.16e-107:
		tmp = x - (-2.0 * (z / t))
	elif z <= 4.8e+101:
		tmp = x - ((z * (y * 2.0)) / t_1)
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * Float64(z * 2.0)) - Float64(y * t))
	t_2 = Float64(x - Float64(y / z))
	tmp = 0.0
	if (z <= -5.5e+154)
		tmp = t_2;
	elseif (z <= -2e-145)
		tmp = Float64(x - Float64(Float64(y * 2.0) / Float64(t_1 / z)));
	elseif (z <= 1.16e-107)
		tmp = Float64(x - Float64(-2.0 * Float64(z / t)));
	elseif (z <= 4.8e+101)
		tmp = Float64(x - Float64(Float64(z * Float64(y * 2.0)) / t_1));
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * (z * 2.0)) - (y * t);
	t_2 = x - (y / z);
	tmp = 0.0;
	if (z <= -5.5e+154)
		tmp = t_2;
	elseif (z <= -2e-145)
		tmp = x - ((y * 2.0) / (t_1 / z));
	elseif (z <= 1.16e-107)
		tmp = x - (-2.0 * (z / t));
	elseif (z <= 4.8e+101)
		tmp = x - ((z * (y * 2.0)) / t_1);
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * N[(z * 2.0), $MachinePrecision]), $MachinePrecision] - N[(y * t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e+154], t$95$2, If[LessEqual[z, -2e-145], N[(x - N[(N[(y * 2.0), $MachinePrecision] / N[(t$95$1 / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.16e-107], N[(x - N[(-2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.8e+101], N[(x - N[(N[(z * N[(y * 2.0), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -5.5000000000000006e154 or 4.79999999999999977e101 < z

   1. Initial program 54.2%

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

      1. sub-neg 54.2%

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

      2. associate-/l* 72.7%

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

      3. distribute-neg-frac 72.7%

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

      4. distribute-lft-neg-out 72.7%

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

      5. associate-/r/ 72.7%

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

      6. distribute-lft-neg-out 72.7%

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

      7. distribute-rgt-neg-in 72.7%

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

      8. metadata-eval 72.7%

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

      9. *-commutative 72.7%

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

      10. associate-*l* 72.7%

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

      11. fma-neg 72.7%

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

   3. Simplified 72.7%

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

   4. Taylor expanded in y around 0 96.7%

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

      1. mul-1-neg 96.7%

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

      2. sub-neg 96.7%

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

   6. Simplified 96.7%

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

   if -5.5000000000000006e154 < z < -1.99999999999999983e-145

   1. Initial program 87.7%

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

      1. remove-double-neg 87.7%

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

      2. neg-mul-1 87.7%

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

      3. *-commutative 87.7%

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

      4. *-commutative 87.7%

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

      5. neg-mul-1 87.7%

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

      6. remove-double-neg 87.7%

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

      7. associate-/l* 94.8%

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

      8. associate-*l* 94.8%

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

   3. Simplified 94.8%

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

   if -1.99999999999999983e-145 < z < 1.16e-107

   1. Initial program 94.4%

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

      1. remove-double-neg 94.4%

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

      2. neg-mul-1 94.4%

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

      3. *-commutative 94.4%

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

      4. *-commutative 94.4%

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

      5. neg-mul-1 94.4%

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

      6. remove-double-neg 94.4%

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

      7. associate-/l* 90.5%

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

      8. associate-*l* 90.5%

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

   3. Simplified 90.5%

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

   4. Taylor expanded in y around inf 98.6%

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

      1. *-commutative 98.6%

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

   6. Simplified 98.6%

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

   if 1.16e-107 < z < 4.79999999999999977e101

   1. Initial program 98.0%

      \[x - \frac{\left(y \cdot 2\right) \cdot z}{\left(z \cdot 2\right) \cdot z - y \cdot t} \]
3. Recombined 4 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+154}:\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{-145}:\\ \;\;\;\;x - \frac{y \cdot 2}{\frac{z \cdot \left(z \cdot 2\right) - y \cdot t}{z}}\\ \mathbf{elif}\;z \leq 1.16 \cdot 10^{-107}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \mathbf{elif}\;z \leq 4.8 \cdot 10^{+101}:\\ \;\;\;\;x - \frac{z \cdot \left(y \cdot 2\right)}{z \cdot \left(z \cdot 2\right) - y \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{z}\\ \end{array} \]

Alternative 5: 89.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-41} \lor \neg \left(z \leq 9.5 \cdot 10^{+20}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -9.5e-41) (not (<= z 9.5e+20)))
   (- x (/ y z))
   (- x (* -2.0 (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.5e-41) || !(z <= 9.5e+20)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (-2.0 * (z / 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 <= (-9.5d-41)) .or. (.not. (z <= 9.5d+20))) then
        tmp = x - (y / z)
    else
        tmp = x - ((-2.0d0) * (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -9.5e-41) || !(z <= 9.5e+20)) {
		tmp = x - (y / z);
	} else {
		tmp = x - (-2.0 * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -9.5e-41) or not (z <= 9.5e+20):
		tmp = x - (y / z)
	else:
		tmp = x - (-2.0 * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -9.5e-41) || !(z <= 9.5e+20))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = Float64(x - Float64(-2.0 * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -9.5e-41) || ~((z <= 9.5e+20)))
		tmp = x - (y / z);
	else
		tmp = x - (-2.0 * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -9.5e-41], N[Not[LessEqual[z, 9.5e+20]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x - N[(-2.0 * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.4999999999999997e-41 or 9.5e20 < z

   1. Initial program 68.1%

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

      1. sub-neg 68.1%

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

      2. associate-/l* 81.7%

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

      3. distribute-neg-frac 81.7%

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

      4. distribute-lft-neg-out 81.7%

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

      5. associate-/r/ 81.1%

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

      6. distribute-lft-neg-out 81.1%

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

      7. distribute-rgt-neg-in 81.1%

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

      8. metadata-eval 81.1%

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

      9. *-commutative 81.1%

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

      10. associate-*l* 81.1%

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

      11. fma-neg 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in y around 0 92.5%

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

      1. mul-1-neg 92.5%

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

      2. sub-neg 92.5%

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

   6. Simplified 92.5%

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

   if -9.4999999999999997e-41 < z < 9.5e20

   1. Initial program 94.3%

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

      1. remove-double-neg 94.3%

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

      2. neg-mul-1 94.3%

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

      3. *-commutative 94.3%

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

      4. *-commutative 94.3%

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

      5. neg-mul-1 94.3%

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

      6. remove-double-neg 94.3%

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

      7. associate-/l* 92.0%

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

      8. associate-*l* 92.0%

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

   3. Simplified 92.0%

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

   4. Taylor expanded in y around inf 92.0%

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

      1. *-commutative 92.0%

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

   6. Simplified 92.0%

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

4. Final simplification 92.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{-41} \lor \neg \left(z \leq 9.5 \cdot 10^{+20}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x - -2 \cdot \frac{z}{t}\\ \end{array} \]

Alternative 6: 82.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-40} \lor \neg \left(z \leq 10^{+21}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.6e-40) (not (<= z 1e+21))) (- x (/ y z)) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.6e-40) || !(z <= 1e+21)) {
		tmp = x - (y / z);
	} else {
		tmp = 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 <= (-3.6d-40)) .or. (.not. (z <= 1d+21))) then
        tmp = x - (y / z)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.6e-40) || !(z <= 1e+21)) {
		tmp = x - (y / z);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.6e-40) or not (z <= 1e+21):
		tmp = x - (y / z)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.6e-40) || !(z <= 1e+21))
		tmp = Float64(x - Float64(y / z));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.6e-40) || ~((z <= 1e+21)))
		tmp = x - (y / z);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.6e-40], N[Not[LessEqual[z, 1e+21]], $MachinePrecision]], N[(x - N[(y / z), $MachinePrecision]), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -3.6e-40 or 1e21 < z

   1. Initial program 68.1%

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

      1. sub-neg 68.1%

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

      2. associate-/l* 81.7%

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

      3. distribute-neg-frac 81.7%

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

      4. distribute-lft-neg-out 81.7%

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

      5. associate-/r/ 81.1%

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

      6. distribute-lft-neg-out 81.1%

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

      7. distribute-rgt-neg-in 81.1%

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

      8. metadata-eval 81.1%

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

      9. *-commutative 81.1%

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

      10. associate-*l* 81.1%

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

      11. fma-neg 81.1%

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

   3. Simplified 81.1%

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

   4. Taylor expanded in y around 0 92.5%

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

      1. mul-1-neg 92.5%

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

      2. sub-neg 92.5%

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

   6. Simplified 92.5%

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

   if -3.6e-40 < z < 1e21

   1. Initial program 94.3%

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

      1. sub-neg 94.3%

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

      2. associate-/l* 92.0%

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

      3. distribute-neg-frac 92.0%

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

      4. distribute-lft-neg-out 92.0%

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

      5. associate-/r/ 95.1%

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

      6. distribute-lft-neg-out 95.1%

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

      7. distribute-rgt-neg-in 95.1%

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

      8. metadata-eval 95.1%

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

      9. *-commutative 95.1%

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

      10. associate-*l* 95.1%

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

      11. fma-neg 95.1%

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

   3. Simplified 95.1%

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

   4. Taylor expanded in x around inf 77.6%

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

4. Final simplification 85.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-40} \lor \neg \left(z \leq 10^{+21}\right):\\ \;\;\;\;x - \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 7: 70.6% accurate, 2.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{-147}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-292}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.08e-147) x (if (<= t -3.1e-292) (/ (- y) z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.08e-147) {
		tmp = x;
	} else if (t <= -3.1e-292) {
		tmp = -y / z;
	} else {
		tmp = 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 (t <= (-1.08d-147)) then
        tmp = x
    else if (t <= (-3.1d-292)) then
        tmp = -y / z
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.08e-147) {
		tmp = x;
	} else if (t <= -3.1e-292) {
		tmp = -y / z;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.08e-147:
		tmp = x
	elif t <= -3.1e-292:
		tmp = -y / z
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.08e-147)
		tmp = x;
	elseif (t <= -3.1e-292)
		tmp = Float64(Float64(-y) / z);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.08e-147)
		tmp = x;
	elseif (t <= -3.1e-292)
		tmp = -y / z;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.08e-147], x, If[LessEqual[t, -3.1e-292], N[((-y) / z), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -1.07999999999999995e-147 or -3.0999999999999999e-292 < t

   1. Initial program 83.5%

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

      1. sub-neg 83.5%

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

      2. associate-/l* 89.9%

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

      3. distribute-neg-frac 89.9%

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

      4. distribute-lft-neg-out 89.9%

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

      5. associate-/r/ 91.1%

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

      6. distribute-lft-neg-out 91.1%

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

      7. distribute-rgt-neg-in 91.1%

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

      8. metadata-eval 91.1%

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

      9. *-commutative 91.1%

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

      10. associate-*l* 91.1%

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

      11. fma-neg 91.1%

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

   3. Simplified 91.1%

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

   4. Taylor expanded in x around inf 79.9%

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

   if -1.07999999999999995e-147 < t < -3.0999999999999999e-292

   1. Initial program 56.1%

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

   3. Simplified 93.3%

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

   4. Taylor expanded in x around 0 66.4%

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

   5. Taylor expanded in y around 0 59.6%

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

      1. mul-1-neg 59.6%

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

      2. distribute-frac-neg 59.6%

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

   7. Simplified 59.6%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.08 \cdot 10^{-147}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq -3.1 \cdot 10^{-292}:\\ \;\;\;\;\frac{-y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 8: 75.2% accurate, 17.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.4%

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

   1. sub-neg 80.4%

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

   2. associate-/l* 86.5%

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

   3. distribute-neg-frac 86.5%

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

   4. distribute-lft-neg-out 86.5%

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

   5. associate-/r/ 87.6%

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

   6. distribute-lft-neg-out 87.6%

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

   7. distribute-rgt-neg-in 87.6%

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

   8. metadata-eval 87.6%

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

   9. *-commutative 87.6%

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

   10. associate-*l* 87.6%

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

   11. fma-neg 87.6%

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

3. Simplified 87.6%

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

4. Taylor expanded in x around inf 74.3%

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

5. Final simplification 74.3%

   \[\leadsto x \]

Developer target: 99.9% accurate, 1.3× 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 2023333 
(FPCore (x y z t)
  :name "Numeric.AD.Rank1.Halley:findZero from ad-4.2.4"
  :precision binary64

  :herbie-target
  (- x (/ 1.0 (- (/ z y) (/ (/ t 2.0) z))))

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