Diagrams.Trail:splitAtParam from diagrams-lib-1.3.0.3, A

Percentage Accurate: 89.0% → 93.7%

Time: 11.3s

Alternatives: 12

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

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 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 89.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 = (x + (((y * z) - x) / ((t * z) - x))) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 93.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -850000000 \lor \neg \left(z \leq 3.65 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{z \cdot t - x}{z \cdot y - x}}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -850000000.0) (not (<= z 3.65e+161)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (/ (+ x (/ 1.0 (/ (- (* z t) x) (- (* z y) x)))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -850000000.0) || !(z <= 3.65e+161)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x + (1.0 / (((z * t) - x) / ((z * y) - x)))) / (x + 1.0);
	}
	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 <= (-850000000.0d0)) .or. (.not. (z <= 3.65d+161))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = (x + (1.0d0 / (((z * t) - x) / ((z * y) - x)))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -850000000.0) || !(z <= 3.65e+161)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x + (1.0 / (((z * t) - x) / ((z * y) - x)))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -850000000.0) or not (z <= 3.65e+161):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = (x + (1.0 / (((z * t) - x) / ((z * y) - x)))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -850000000.0) || !(z <= 3.65e+161))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + Float64(1.0 / Float64(Float64(Float64(z * t) - x) / Float64(Float64(z * y) - x)))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -850000000.0) || ~((z <= 3.65e+161)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = (x + (1.0 / (((z * t) - x) / ((z * y) - x)))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -850000000.0], N[Not[LessEqual[z, 3.65e+161]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 / N[(N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e8 or 3.6499999999999998e161 < z

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in z around inf 90.8%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

   if -8.5e8 < z < 3.6499999999999998e161

   1. Initial program 97.1%

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

      1. *-commutative 97.1%

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

   3. Simplified 97.1%

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

      1. clear-num 97.1%

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

      2. inv-pow 97.1%

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

      3. fma-neg 97.1%

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

   5. Applied egg-rr 97.1%

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

      1. unpow-1 97.1%

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

      2. *-commutative 97.1%

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

      3. fma-neg 97.1%

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

      4. *-commutative 97.1%

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

   7. Simplified 97.1%

      \[\leadsto \frac{x + \color{blue}{\frac{1}{\frac{t \cdot z - x}{z \cdot y - x}}}}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -850000000 \lor \neg \left(z \leq 3.65 \cdot 10^{+161}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{z \cdot t - x}{z \cdot y - x}}}{x + 1}\\ \end{array} \]

Alternative 2: 93.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -850000000 \lor \neg \left(z \leq 8.2 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -850000000.0) (not (<= z 8.2e+160)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (/ (+ x (/ (- (* z y) x) (- (* z t) x))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -850000000.0) || !(z <= 8.2e+160)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	}
	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 <= (-850000000.0d0)) .or. (.not. (z <= 8.2d+160))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -850000000.0) || !(z <= 8.2e+160)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -850000000.0) or not (z <= 8.2e+160):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -850000000.0) || !(z <= 8.2e+160))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x + Float64(Float64(Float64(z * y) - x) / Float64(Float64(z * t) - x))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -850000000.0) || ~((z <= 8.2e+160)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = (x + (((z * y) - x) / ((z * t) - x))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -850000000.0], N[Not[LessEqual[z, 8.2e+160]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(N[(N[(z * y), $MachinePrecision] - x), $MachinePrecision] / N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e8 or 8.19999999999999996e160 < z

   1. Initial program 70.8%

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

      1. *-commutative 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in z around inf 90.8%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

   if -8.5e8 < z < 8.19999999999999996e160

   1. Initial program 97.1%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -850000000 \lor \neg \left(z \leq 8.2 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{z \cdot y - x}{z \cdot t - x}}{x + 1}\\ \end{array} \]

Alternative 3: 76.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{y}{t}}{x + 1}\\ t_2 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -0.0016:\\ \;\;\;\;t_2\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-89}:\\ \;\;\;\;1 + z \cdot \left(y - \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-124}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-144}:\\ \;\;\;\;z \cdot y + \left(1 - \frac{y}{\frac{x}{z}}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ y t)) (+ x 1.0))) (t_2 (/ x (+ x 1.0))))
   (if (<= x -0.0016)
     t_2
     (if (<= x -4.8e-89)
       (+ 1.0 (* z (- y (/ y x))))
       (if (<= x -1.16e-124)
         t_1
         (if (<= x -8.6e-144)
           (+ (* z y) (- 1.0 (/ y (/ x z))))
           (if (<= x 1.35e+42) t_1 t_2)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + (y / t)) / (x + 1.0);
	double t_2 = x / (x + 1.0);
	double tmp;
	if (x <= -0.0016) {
		tmp = t_2;
	} else if (x <= -4.8e-89) {
		tmp = 1.0 + (z * (y - (y / x)));
	} else if (x <= -1.16e-124) {
		tmp = t_1;
	} else if (x <= -8.6e-144) {
		tmp = (z * y) + (1.0 - (y / (x / z)));
	} else if (x <= 1.35e+42) {
		tmp = 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 = (x + (y / t)) / (x + 1.0d0)
    t_2 = x / (x + 1.0d0)
    if (x <= (-0.0016d0)) then
        tmp = t_2
    else if (x <= (-4.8d-89)) then
        tmp = 1.0d0 + (z * (y - (y / x)))
    else if (x <= (-1.16d-124)) then
        tmp = t_1
    else if (x <= (-8.6d-144)) then
        tmp = (z * y) + (1.0d0 - (y / (x / z)))
    else if (x <= 1.35d+42) then
        tmp = 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 = (x + (y / t)) / (x + 1.0);
	double t_2 = x / (x + 1.0);
	double tmp;
	if (x <= -0.0016) {
		tmp = t_2;
	} else if (x <= -4.8e-89) {
		tmp = 1.0 + (z * (y - (y / x)));
	} else if (x <= -1.16e-124) {
		tmp = t_1;
	} else if (x <= -8.6e-144) {
		tmp = (z * y) + (1.0 - (y / (x / z)));
	} else if (x <= 1.35e+42) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + (y / t)) / (x + 1.0)
	t_2 = x / (x + 1.0)
	tmp = 0
	if x <= -0.0016:
		tmp = t_2
	elif x <= -4.8e-89:
		tmp = 1.0 + (z * (y - (y / x)))
	elif x <= -1.16e-124:
		tmp = t_1
	elif x <= -8.6e-144:
		tmp = (z * y) + (1.0 - (y / (x / z)))
	elif x <= 1.35e+42:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0))
	t_2 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -0.0016)
		tmp = t_2;
	elseif (x <= -4.8e-89)
		tmp = Float64(1.0 + Float64(z * Float64(y - Float64(y / x))));
	elseif (x <= -1.16e-124)
		tmp = t_1;
	elseif (x <= -8.6e-144)
		tmp = Float64(Float64(z * y) + Float64(1.0 - Float64(y / Float64(x / z))));
	elseif (x <= 1.35e+42)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + (y / t)) / (x + 1.0);
	t_2 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -0.0016)
		tmp = t_2;
	elseif (x <= -4.8e-89)
		tmp = 1.0 + (z * (y - (y / x)));
	elseif (x <= -1.16e-124)
		tmp = t_1;
	elseif (x <= -8.6e-144)
		tmp = (z * y) + (1.0 - (y / (x / z)));
	elseif (x <= 1.35e+42)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0016], t$95$2, If[LessEqual[x, -4.8e-89], N[(1.0 + N[(z * N[(y - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, -1.16e-124], t$95$1, If[LessEqual[x, -8.6e-144], N[(N[(z * y), $MachinePrecision] + N[(1.0 - N[(y / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e+42], t$95$1, t$95$2]]]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -0.00160000000000000008 or 1.35e42 < x

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in t around inf 87.7%

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

      1. +-commutative 87.7%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   6. Simplified 87.7%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if -0.00160000000000000008 < x < -4.80000000000000032e-89

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 72.2%

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

      1. fma-neg 72.2%

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

      2. associate-*r/ 72.2%

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

      3. neg-mul-1 72.2%

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

      4. neg-sub0 72.2%

         \[\leadsto \frac{x + \frac{\color{blue}{0 - \mathsf{fma}\left(y, z, -x\right)}}{x}}{x + 1} \]

      5. fma-def 72.2%

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

      6. +-commutative 72.2%

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

      7. associate--r+ 72.2%

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

      8. neg-sub0 72.2%

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

      9. remove-double-neg 72.2%

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

      10. *-commutative 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in x around 0 72.2%

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

      1. mul-1-neg 72.2%

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

      2. *-commutative 72.2%

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

      3. sub-neg 72.2%

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

      4. associate-*r* 72.2%

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

      5. neg-mul-1 72.2%

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

      6. cancel-sign-sub 72.2%

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

      7. *-commutative 72.2%

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

      8. associate-/l* 72.2%

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

   9. Simplified 72.2%

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

   10. Taylor expanded in z around 0 72.2%

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

   if -4.80000000000000032e-89 < x < -1.1600000000000001e-124 or -8.5999999999999998e-144 < x < 1.35e42

   1. Initial program 88.3%

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

      1. *-commutative 88.3%

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

   3. Simplified 88.3%

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

   4. Taylor expanded in z around inf 78.2%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

   if -1.1600000000000001e-124 < x < -8.5999999999999998e-144

   1. Initial program 99.7%

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

      1. *-commutative 99.7%

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

   3. Simplified 99.7%

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

   4. Taylor expanded in t around 0 95.3%

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

      1. fma-neg 95.3%

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

      2. associate-*r/ 95.3%

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

      3. neg-mul-1 95.3%

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

      4. neg-sub0 95.3%

         \[\leadsto \frac{x + \frac{\color{blue}{0 - \mathsf{fma}\left(y, z, -x\right)}}{x}}{x + 1} \]

      5. fma-def 95.3%

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

      6. +-commutative 95.3%

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

      7. associate--r+ 95.3%

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

      8. neg-sub0 95.3%

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

      9. remove-double-neg 95.3%

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

      10. *-commutative 95.3%

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

   6. Simplified 95.3%

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

   7. Taylor expanded in x around 0 95.3%

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

      1. mul-1-neg 95.3%

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

      2. *-commutative 95.3%

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

      3. sub-neg 95.3%

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

      4. associate-*r* 95.3%

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

      5. neg-mul-1 95.3%

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

      6. cancel-sign-sub 95.3%

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

      7. *-commutative 95.3%

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

      8. associate-/l* 95.6%

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

   9. Simplified 95.6%

      \[\leadsto \color{blue}{\left(1 - \frac{y}{\frac{x}{z}}\right) + y \cdot z} \]
3. Recombined 4 regimes into one program.

4. Final simplification 82.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0016:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-89}:\\ \;\;\;\;1 + z \cdot \left(y - \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq -1.16 \cdot 10^{-124}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{elif}\;x \leq -8.6 \cdot 10^{-144}:\\ \;\;\;\;z \cdot y + \left(1 - \frac{y}{\frac{x}{z}}\right)\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{+42}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 4: 82.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-116} \lor \neg \left(t \leq 3.5 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - \frac{y}{\frac{x}{z}}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4e-116) (not (<= t 3.5e-16)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (/ (- (+ x 1.0) (/ y (/ x z))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4e-116) || !(t <= 3.5e-16)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = ((x + 1.0) - (y / (x / z))) / (x + 1.0);
	}
	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 <= (-4d-116)) .or. (.not. (t <= 3.5d-16))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = ((x + 1.0d0) - (y / (x / z))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4e-116) || !(t <= 3.5e-16)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = ((x + 1.0) - (y / (x / z))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -4e-116) or not (t <= 3.5e-16):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = ((x + 1.0) - (y / (x / z))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4e-116) || !(t <= 3.5e-16))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(Float64(x + 1.0) - Float64(y / Float64(x / z))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4e-116) || ~((t <= 3.5e-16)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = ((x + 1.0) - (y / (x / z))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4e-116], N[Not[LessEqual[t, 3.5e-16]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(N[(x + 1.0), $MachinePrecision] - N[(y / N[(x / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4e-116 or 3.50000000000000017e-16 < t

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in z around inf 86.9%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

   if -4e-116 < t < 3.50000000000000017e-16

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in t around 0 78.0%

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

      1. associate-+r+ 78.0%

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

      2. mul-1-neg 78.0%

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

      3. unsub-neg 78.0%

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

      4. +-commutative 78.0%

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

      5. associate-/l* 84.8%

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

      6. +-commutative 84.8%

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

   6. Simplified 84.8%

      \[\leadsto \color{blue}{\frac{\left(x + 1\right) - \frac{y}{\frac{x}{z}}}{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 86.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4 \cdot 10^{-116} \lor \neg \left(t \leq 3.5 \cdot 10^{-16}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x + 1\right) - \frac{y}{\frac{x}{z}}}{x + 1}\\ \end{array} \]

Alternative 5: 76.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -0.0225:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-89}:\\ \;\;\;\;1 + z \cdot \left(y - \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+42}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (+ x 1.0))))
   (if (<= x -0.0225)
     t_1
     (if (<= x -4.8e-89)
       (+ 1.0 (* z (- y (/ y x))))
       (if (<= x 2.75e+42) (/ (+ x (/ y t)) (+ x 1.0)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -0.0225) {
		tmp = t_1;
	} else if (x <= -4.8e-89) {
		tmp = 1.0 + (z * (y - (y / x)));
	} else if (x <= 2.75e+42) {
		tmp = (x + (y / t)) / (x + 1.0);
	} 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 / (x + 1.0d0)
    if (x <= (-0.0225d0)) then
        tmp = t_1
    else if (x <= (-4.8d-89)) then
        tmp = 1.0d0 + (z * (y - (y / x)))
    else if (x <= 2.75d+42) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    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 / (x + 1.0);
	double tmp;
	if (x <= -0.0225) {
		tmp = t_1;
	} else if (x <= -4.8e-89) {
		tmp = 1.0 + (z * (y - (y / x)));
	} else if (x <= 2.75e+42) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -0.0225:
		tmp = t_1
	elif x <= -4.8e-89:
		tmp = 1.0 + (z * (y - (y / x)))
	elif x <= 2.75e+42:
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -0.0225)
		tmp = t_1;
	elseif (x <= -4.8e-89)
		tmp = Float64(1.0 + Float64(z * Float64(y - Float64(y / x))));
	elseif (x <= 2.75e+42)
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -0.0225)
		tmp = t_1;
	elseif (x <= -4.8e-89)
		tmp = 1.0 + (z * (y - (y / x)));
	elseif (x <= 2.75e+42)
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.0225], t$95$1, If[LessEqual[x, -4.8e-89], N[(1.0 + N[(z * N[(y - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 2.75e+42], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -0.022499999999999999 or 2.75000000000000001e42 < x

   1. Initial program 85.4%

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

      1. *-commutative 85.4%

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

   3. Simplified 85.4%

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

   4. Taylor expanded in t around inf 87.7%

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

      1. +-commutative 87.7%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   6. Simplified 87.7%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if -0.022499999999999999 < x < -4.80000000000000032e-89

   1. Initial program 100.0%

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

      1. *-commutative 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in t around 0 72.2%

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

      1. fma-neg 72.2%

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

      2. associate-*r/ 72.2%

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

      3. neg-mul-1 72.2%

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

      4. neg-sub0 72.2%

         \[\leadsto \frac{x + \frac{\color{blue}{0 - \mathsf{fma}\left(y, z, -x\right)}}{x}}{x + 1} \]

      5. fma-def 72.2%

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

      6. +-commutative 72.2%

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

      7. associate--r+ 72.2%

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

      8. neg-sub0 72.2%

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

      9. remove-double-neg 72.2%

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

      10. *-commutative 72.2%

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

   6. Simplified 72.2%

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

   7. Taylor expanded in x around 0 72.2%

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

      1. mul-1-neg 72.2%

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

      2. *-commutative 72.2%

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

      3. sub-neg 72.2%

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

      4. associate-*r* 72.2%

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

      5. neg-mul-1 72.2%

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

      6. cancel-sign-sub 72.2%

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

      7. *-commutative 72.2%

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

      8. associate-/l* 72.2%

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

   9. Simplified 72.2%

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

   10. Taylor expanded in z around 0 72.2%

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

   if -4.80000000000000032e-89 < x < 2.75000000000000001e42

   1. Initial program 88.8%

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

      1. *-commutative 88.8%

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

   3. Simplified 88.8%

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

   4. Taylor expanded in z around inf 75.4%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 3 regimes into one program.

4. Final simplification 80.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.0225:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -4.8 \cdot 10^{-89}:\\ \;\;\;\;1 + z \cdot \left(y - \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 2.75 \cdot 10^{+42}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 6: 81.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-116} \lor \neg \left(t \leq 1.5 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{z}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -5.8e-116) (not (<= t 1.5e-18)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (- 1.0 (* (/ y x) (/ z (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.8e-116) || !(t <= 1.5e-18)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - ((y / x) * (z / (x + 1.0)));
	}
	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 <= (-5.8d-116)) .or. (.not. (t <= 1.5d-18))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 1.0d0 - ((y / x) * (z / (x + 1.0d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.8e-116) || !(t <= 1.5e-18)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0 - ((y / x) * (z / (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -5.8e-116) or not (t <= 1.5e-18):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0 - ((y / x) * (z / (x + 1.0)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -5.8e-116) || !(t <= 1.5e-18))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(1.0 - Float64(Float64(y / x) * Float64(z / Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -5.8e-116) || ~((t <= 1.5e-18)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0 - ((y / x) * (z / (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -5.8e-116], N[Not[LessEqual[t, 1.5e-18]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(y / x), $MachinePrecision] * N[(z / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.7999999999999996e-116 or 1.49999999999999991e-18 < t

   1. Initial program 87.0%

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

      1. *-commutative 87.0%

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

   3. Simplified 87.0%

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

   4. Taylor expanded in z around inf 86.9%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]

   if -5.7999999999999996e-116 < t < 1.49999999999999991e-18

   1. Initial program 90.2%

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

      1. *-commutative 90.2%

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

   3. Simplified 90.2%

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

   4. Taylor expanded in t around 0 78.0%

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

      1. fma-neg 78.0%

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

      2. associate-*r/ 78.0%

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

      3. neg-mul-1 78.0%

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

      4. neg-sub0 78.0%

         \[\leadsto \frac{x + \frac{\color{blue}{0 - \mathsf{fma}\left(y, z, -x\right)}}{x}}{x + 1} \]

      5. fma-def 78.0%

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

      6. +-commutative 78.0%

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

      7. associate--r+ 78.0%

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

      8. neg-sub0 78.0%

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

      9. remove-double-neg 78.0%

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

      10. *-commutative 78.0%

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

   6. Simplified 78.0%

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

   7. Taylor expanded in z around 0 77.9%

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

      1. mul-1-neg 77.9%

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

      2. sub-neg 77.9%

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

      3. times-frac 83.9%

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

      4. +-commutative 83.9%

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

   9. Simplified 83.9%

      \[\leadsto \color{blue}{1 - \frac{y}{x} \cdot \frac{z}{x + 1}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.8 \cdot 10^{-116} \lor \neg \left(t \leq 1.5 \cdot 10^{-18}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{y}{x} \cdot \frac{z}{x + 1}\\ \end{array} \]

Alternative 7: 68.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{x + 1}\\ \mathbf{if}\;x \leq -0.000445:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-144}:\\ \;\;\;\;1 + z \cdot \left(y - \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (+ x 1.0))))
   (if (<= x -0.000445)
     t_1
     (if (<= x -4.9e-144)
       (+ 1.0 (* z (- y (/ y x))))
       (if (<= x 5.8e-95) (/ y t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x / (x + 1.0);
	double tmp;
	if (x <= -0.000445) {
		tmp = t_1;
	} else if (x <= -4.9e-144) {
		tmp = 1.0 + (z * (y - (y / x)));
	} else if (x <= 5.8e-95) {
		tmp = y / 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 = x / (x + 1.0d0)
    if (x <= (-0.000445d0)) then
        tmp = t_1
    else if (x <= (-4.9d-144)) then
        tmp = 1.0d0 + (z * (y - (y / x)))
    else if (x <= 5.8d-95) then
        tmp = y / 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 = x / (x + 1.0);
	double tmp;
	if (x <= -0.000445) {
		tmp = t_1;
	} else if (x <= -4.9e-144) {
		tmp = 1.0 + (z * (y - (y / x)));
	} else if (x <= 5.8e-95) {
		tmp = y / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x / (x + 1.0)
	tmp = 0
	if x <= -0.000445:
		tmp = t_1
	elif x <= -4.9e-144:
		tmp = 1.0 + (z * (y - (y / x)))
	elif x <= 5.8e-95:
		tmp = y / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -0.000445)
		tmp = t_1;
	elseif (x <= -4.9e-144)
		tmp = Float64(1.0 + Float64(z * Float64(y - Float64(y / x))));
	elseif (x <= 5.8e-95)
		tmp = Float64(y / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x / (x + 1.0);
	tmp = 0.0;
	if (x <= -0.000445)
		tmp = t_1;
	elseif (x <= -4.9e-144)
		tmp = 1.0 + (z * (y - (y / x)));
	elseif (x <= 5.8e-95)
		tmp = y / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -0.000445], t$95$1, If[LessEqual[x, -4.9e-144], N[(1.0 + N[(z * N[(y - N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5.8e-95], N[(y / t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.4499999999999997e-4 or 5.80000000000000004e-95 < x

   1. Initial program 84.4%

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

      1. *-commutative 84.4%

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

   3. Simplified 84.4%

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

   4. Taylor expanded in t around inf 81.1%

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

      1. +-commutative 81.1%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   6. Simplified 81.1%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   if -4.4499999999999997e-4 < x < -4.9000000000000001e-144

   1. Initial program 97.1%

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

      1. *-commutative 97.1%

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

   3. Simplified 97.1%

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

   4. Taylor expanded in t around 0 63.3%

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

      1. fma-neg 63.3%

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

      2. associate-*r/ 63.3%

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

      3. neg-mul-1 63.3%

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

      4. neg-sub0 63.3%

         \[\leadsto \frac{x + \frac{\color{blue}{0 - \mathsf{fma}\left(y, z, -x\right)}}{x}}{x + 1} \]

      5. fma-def 63.3%

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

      6. +-commutative 63.3%

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

      7. associate--r+ 63.3%

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

      8. neg-sub0 63.3%

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

      9. remove-double-neg 63.3%

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

      10. *-commutative 63.3%

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

   6. Simplified 63.3%

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

   7. Taylor expanded in x around 0 63.3%

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

      1. mul-1-neg 63.3%

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

      2. *-commutative 63.3%

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

      3. sub-neg 63.3%

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

      4. associate-*r* 63.3%

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

      5. neg-mul-1 63.3%

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

      6. cancel-sign-sub 63.3%

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

      7. *-commutative 63.3%

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

      8. associate-/l* 63.3%

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

   9. Simplified 63.3%

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

   10. Taylor expanded in z around 0 60.5%

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

   if -4.9000000000000001e-144 < x < 5.80000000000000004e-95

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

   3. Simplified 90.7%

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

      1. clear-num 90.8%

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

      2. inv-pow 90.8%

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

      3. fma-neg 90.8%

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

   5. Applied egg-rr 90.8%

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

      1. unpow-1 90.8%

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

      2. *-commutative 90.8%

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

      3. fma-neg 90.8%

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

      4. *-commutative 90.8%

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

   7. Simplified 90.8%

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

   8. Taylor expanded in x around 0 57.7%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000445:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq -4.9 \cdot 10^{-144}:\\ \;\;\;\;1 + z \cdot \left(y - \frac{y}{x}\right)\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 8: 67.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{1}{x}\\ \mathbf{if}\;x \leq -2.85 \cdot 10^{-89}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ 1.0 x))))
   (if (<= x -2.85e-89)
     t_1
     (if (<= x 3.1e-94) (/ y t) (if (<= x 3.3e-29) x t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (1.0 / x);
	double tmp;
	if (x <= -2.85e-89) {
		tmp = t_1;
	} else if (x <= 3.1e-94) {
		tmp = y / t;
	} else if (x <= 3.3e-29) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (1.0d0 / x)
    if (x <= (-2.85d-89)) then
        tmp = t_1
    else if (x <= 3.1d-94) then
        tmp = y / t
    else if (x <= 3.3d-29) then
        tmp = x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (1.0 / x);
	double tmp;
	if (x <= -2.85e-89) {
		tmp = t_1;
	} else if (x <= 3.1e-94) {
		tmp = y / t;
	} else if (x <= 3.3e-29) {
		tmp = x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (1.0 / x)
	tmp = 0
	if x <= -2.85e-89:
		tmp = t_1
	elif x <= 3.1e-94:
		tmp = y / t
	elif x <= 3.3e-29:
		tmp = x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(1.0 / x))
	tmp = 0.0
	if (x <= -2.85e-89)
		tmp = t_1;
	elseif (x <= 3.1e-94)
		tmp = Float64(y / t);
	elseif (x <= 3.3e-29)
		tmp = x;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (1.0 / x);
	tmp = 0.0;
	if (x <= -2.85e-89)
		tmp = t_1;
	elseif (x <= 3.1e-94)
		tmp = y / t;
	elseif (x <= 3.3e-29)
		tmp = x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(1.0 / x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -2.85e-89], t$95$1, If[LessEqual[x, 3.1e-94], N[(y / t), $MachinePrecision], If[LessEqual[x, 3.3e-29], x, t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8500000000000001e-89 or 3.30000000000000028e-29 < x

   1. Initial program 86.7%

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

      1. *-commutative 86.7%

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

   3. Simplified 86.7%

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

   4. Taylor expanded in t around inf 75.9%

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

      1. +-commutative 75.9%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   6. Simplified 75.9%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]
   7. Step-by-step derivation

      1. div-inv 75.8%

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

   8. Applied egg-rr 75.8%

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

   9. Taylor expanded in x around inf 79.3%

      \[\leadsto x \cdot \color{blue}{\frac{1}{x}} \]

   if -2.8500000000000001e-89 < x < 3.0999999999999998e-94

   1. Initial program 91.0%

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

      1. *-commutative 91.0%

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

   3. Simplified 91.0%

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

      1. clear-num 91.0%

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

      2. inv-pow 91.0%

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

      3. fma-neg 91.0%

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

   5. Applied egg-rr 91.0%

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

      1. unpow-1 91.0%

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

      2. *-commutative 91.0%

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

      3. fma-neg 91.0%

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

      4. *-commutative 91.0%

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

   7. Simplified 91.0%

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

   8. Taylor expanded in x around 0 54.1%

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

   if 3.0999999999999998e-94 < x < 3.30000000000000028e-29

   1. Initial program 84.6%

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

      1. *-commutative 84.6%

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

   3. Simplified 84.6%

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

   4. Taylor expanded in t around inf 55.6%

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

      1. +-commutative 55.6%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   6. Simplified 55.6%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   7. Taylor expanded in x around 0 55.6%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 3.1 \cdot 10^{-94}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{elif}\;x \leq 3.3 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{x}\\ \end{array} \]

Alternative 9: 67.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.85e-89)
   (* x (/ 1.0 x))
   (if (<= x 7.5e-92) (/ y t) (/ x (+ x 1.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.85e-89) {
		tmp = x * (1.0 / x);
	} else if (x <= 7.5e-92) {
		tmp = y / t;
	} else {
		tmp = x / (x + 1.0);
	}
	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 <= (-2.85d-89)) then
        tmp = x * (1.0d0 / x)
    else if (x <= 7.5d-92) then
        tmp = y / t
    else
        tmp = x / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.85e-89) {
		tmp = x * (1.0 / x);
	} else if (x <= 7.5e-92) {
		tmp = y / t;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.85e-89:
		tmp = x * (1.0 / x)
	elif x <= 7.5e-92:
		tmp = y / t
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.85e-89)
		tmp = Float64(x * Float64(1.0 / x));
	elseif (x <= 7.5e-92)
		tmp = Float64(y / t);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.85e-89)
		tmp = x * (1.0 / x);
	elseif (x <= 7.5e-92)
		tmp = y / t;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.85e-89], N[(x * N[(1.0 / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.5e-92], N[(y / t), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8500000000000001e-89

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in t around inf 74.8%

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

      1. +-commutative 74.8%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   6. Simplified 74.8%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]
   7. Step-by-step derivation

      1. div-inv 74.7%

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

   8. Applied egg-rr 74.7%

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

   9. Taylor expanded in x around inf 79.0%

      \[\leadsto x \cdot \color{blue}{\frac{1}{x}} \]

   if -2.8500000000000001e-89 < x < 7.5000000000000005e-92

   1. Initial program 91.0%

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

      1. *-commutative 91.0%

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

   3. Simplified 91.0%

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

      1. clear-num 91.0%

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

      2. inv-pow 91.0%

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

      3. fma-neg 91.0%

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

   5. Applied egg-rr 91.0%

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

      1. unpow-1 91.0%

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

      2. *-commutative 91.0%

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

      3. fma-neg 91.0%

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

      4. *-commutative 91.0%

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

   7. Simplified 91.0%

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

   8. Taylor expanded in x around 0 54.1%

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

   if 7.5000000000000005e-92 < x

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in t around inf 73.6%

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

      1. +-commutative 73.6%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   6. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 67.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-89}:\\ \;\;\;\;x \cdot \frac{1}{x}\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-92}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 10: 67.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-89}:\\ \;\;\;\;\frac{x + 1}{x + 1}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.85e-89)
   (/ (+ x 1.0) (+ x 1.0))
   (if (<= x 1.35e-95) (/ y t) (/ x (+ x 1.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.85e-89) {
		tmp = (x + 1.0) / (x + 1.0);
	} else if (x <= 1.35e-95) {
		tmp = y / t;
	} else {
		tmp = x / (x + 1.0);
	}
	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 <= (-2.85d-89)) then
        tmp = (x + 1.0d0) / (x + 1.0d0)
    else if (x <= 1.35d-95) then
        tmp = y / t
    else
        tmp = x / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.85e-89) {
		tmp = (x + 1.0) / (x + 1.0);
	} else if (x <= 1.35e-95) {
		tmp = y / t;
	} else {
		tmp = x / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.85e-89:
		tmp = (x + 1.0) / (x + 1.0)
	elif x <= 1.35e-95:
		tmp = y / t
	else:
		tmp = x / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.85e-89)
		tmp = Float64(Float64(x + 1.0) / Float64(x + 1.0));
	elseif (x <= 1.35e-95)
		tmp = Float64(y / t);
	else
		tmp = Float64(x / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.85e-89)
		tmp = (x + 1.0) / (x + 1.0);
	elseif (x <= 1.35e-95)
		tmp = y / t;
	else
		tmp = x / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.85e-89], N[(N[(x + 1.0), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.35e-95], N[(y / t), $MachinePrecision], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.8500000000000001e-89

   1. Initial program 90.7%

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

      1. *-commutative 90.7%

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

   3. Simplified 90.7%

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

   4. Taylor expanded in z around 0 79.1%

      \[\leadsto \frac{x + \color{blue}{1}}{x + 1} \]

   if -2.8500000000000001e-89 < x < 1.35e-95

   1. Initial program 91.0%

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

      1. *-commutative 91.0%

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

   3. Simplified 91.0%

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

      1. clear-num 91.0%

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

      2. inv-pow 91.0%

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

      3. fma-neg 91.0%

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

   5. Applied egg-rr 91.0%

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

      1. unpow-1 91.0%

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

      2. *-commutative 91.0%

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

      3. fma-neg 91.0%

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

      4. *-commutative 91.0%

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

   7. Simplified 91.0%

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

   8. Taylor expanded in x around 0 54.1%

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

   if 1.35e-95 < x

   1. Initial program 81.8%

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

      1. *-commutative 81.8%

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

   3. Simplified 81.8%

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

   4. Taylor expanded in t around inf 73.6%

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

      1. +-commutative 73.6%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   6. Simplified 73.6%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 68.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.85 \cdot 10^{-89}:\\ \;\;\;\;\frac{x + 1}{x + 1}\\ \mathbf{elif}\;x \leq 1.35 \cdot 10^{-95}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1}\\ \end{array} \]

Alternative 11: 25.9% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+63} \lor \neg \left(y \leq 3.05 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.55e+63) (not (<= y 3.05e-31))) (/ y t) x))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.55e+63) || !(y <= 3.05e-31)) {
		tmp = y / t;
	} 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 ((y <= (-1.55d+63)) .or. (.not. (y <= 3.05d-31))) then
        tmp = y / t
    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 ((y <= -1.55e+63) || !(y <= 3.05e-31)) {
		tmp = y / t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.55e+63) or not (y <= 3.05e-31):
		tmp = y / t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.55e+63) || !(y <= 3.05e-31))
		tmp = Float64(y / t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.55e+63) || ~((y <= 3.05e-31)))
		tmp = y / t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.55e+63], N[Not[LessEqual[y, 3.05e-31]], $MachinePrecision]], N[(y / t), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if y < -1.55e63 or 3.0499999999999999e-31 < y

   1. Initial program 77.3%

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

      1. *-commutative 77.3%

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

   3. Simplified 77.3%

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

      1. clear-num 77.2%

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

      2. inv-pow 77.2%

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

      3. fma-neg 77.2%

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

   5. Applied egg-rr 77.2%

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

      1. unpow-1 77.2%

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

      2. *-commutative 77.2%

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

      3. fma-neg 77.2%

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

      4. *-commutative 77.2%

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

   7. Simplified 77.2%

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

   8. Taylor expanded in x around 0 33.1%

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

   if -1.55e63 < y < 3.0499999999999999e-31

   1. Initial program 99.2%

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

      1. *-commutative 99.2%

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

   3. Simplified 99.2%

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

   4. Taylor expanded in t around inf 69.9%

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

      1. +-commutative 69.9%

         \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

   6. Simplified 69.9%

      \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

   7. Taylor expanded in x around 0 28.5%

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

4. Final simplification 30.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.55 \cdot 10^{+63} \lor \neg \left(y \leq 3.05 \cdot 10^{-31}\right):\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]

Alternative 12: 12.8% 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 88.2%

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

   1. *-commutative 88.2%

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

3. Simplified 88.2%

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

4. Taylor expanded in t around inf 55.8%

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

   1. +-commutative 55.8%

      \[\leadsto \frac{x}{\color{blue}{x + 1}} \]

6. Simplified 55.8%

   \[\leadsto \color{blue}{\frac{x}{x + 1}} \]

7. Taylor expanded in x around 0 16.7%

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

8. Final simplification 16.7%

   \[\leadsto x \]

Developer target: 99.4% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

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 = (x + ((y / (t - (x / z))) - (x / ((t * z) - x)))) / (x + 1.0d0)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023309 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

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

  (/ (+ x (/ (- (* y z) x) (- (* t z) x))) (+ x 1.0)))