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

Percentage Accurate: 88.8% → 96.6%

Time: 9.8s

Alternatives: 10

Speedup: 0.7×

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: 88.8% 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: 96.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot z - x\\ t_2 := z \cdot t - x\\ t_3 := \frac{x + \frac{t_1}{t_2}}{x + 1}\\ \mathbf{if}\;t_3 \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{\frac{\left(x + 1\right) \cdot t_2}{z}}\\ \mathbf{elif}\;t_3 \leq 2 \cdot 10^{-27} \lor \neg \left(t_3 \leq 10^{+253}\right):\\ \;\;\;\;\left(\frac{y}{t \cdot \left(x + 1\right)} + \frac{x}{x + 1}\right) - \frac{x}{\left(x + 1\right) \cdot \left(z \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{t_2}{t_1}}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* y z) x))
        (t_2 (- (* z t) x))
        (t_3 (/ (+ x (/ t_1 t_2)) (+ x 1.0))))
   (if (<= t_3 -1e+25)
     (/ y (/ (* (+ x 1.0) t_2) z))
     (if (or (<= t_3 2e-27) (not (<= t_3 1e+253)))
       (-
        (+ (/ y (* t (+ x 1.0))) (/ x (+ x 1.0)))
        (/ x (* (+ x 1.0) (* z t))))
       (/ (+ x (/ 1.0 (/ t_2 t_1))) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - x;
	double t_2 = (z * t) - x;
	double t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1e+25) {
		tmp = y / (((x + 1.0) * t_2) / z);
	} else if ((t_3 <= 2e-27) || !(t_3 <= 1e+253)) {
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / ((x + 1.0) * (z * t)));
	} else {
		tmp = (x + (1.0 / (t_2 / t_1))) / (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) :: t_1
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_1 = (y * z) - x
    t_2 = (z * t) - x
    t_3 = (x + (t_1 / t_2)) / (x + 1.0d0)
    if (t_3 <= (-1d+25)) then
        tmp = y / (((x + 1.0d0) * t_2) / z)
    else if ((t_3 <= 2d-27) .or. (.not. (t_3 <= 1d+253))) then
        tmp = ((y / (t * (x + 1.0d0))) + (x / (x + 1.0d0))) - (x / ((x + 1.0d0) * (z * t)))
    else
        tmp = (x + (1.0d0 / (t_2 / t_1))) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y * z) - x;
	double t_2 = (z * t) - x;
	double t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	double tmp;
	if (t_3 <= -1e+25) {
		tmp = y / (((x + 1.0) * t_2) / z);
	} else if ((t_3 <= 2e-27) || !(t_3 <= 1e+253)) {
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / ((x + 1.0) * (z * t)));
	} else {
		tmp = (x + (1.0 / (t_2 / t_1))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y * z) - x
	t_2 = (z * t) - x
	t_3 = (x + (t_1 / t_2)) / (x + 1.0)
	tmp = 0
	if t_3 <= -1e+25:
		tmp = y / (((x + 1.0) * t_2) / z)
	elif (t_3 <= 2e-27) or not (t_3 <= 1e+253):
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / ((x + 1.0) * (z * t)))
	else:
		tmp = (x + (1.0 / (t_2 / t_1))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y * z) - x)
	t_2 = Float64(Float64(z * t) - x)
	t_3 = Float64(Float64(x + Float64(t_1 / t_2)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_3 <= -1e+25)
		tmp = Float64(y / Float64(Float64(Float64(x + 1.0) * t_2) / z));
	elseif ((t_3 <= 2e-27) || !(t_3 <= 1e+253))
		tmp = Float64(Float64(Float64(y / Float64(t * Float64(x + 1.0))) + Float64(x / Float64(x + 1.0))) - Float64(x / Float64(Float64(x + 1.0) * Float64(z * t))));
	else
		tmp = Float64(Float64(x + Float64(1.0 / Float64(t_2 / t_1))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y * z) - x;
	t_2 = (z * t) - x;
	t_3 = (x + (t_1 / t_2)) / (x + 1.0);
	tmp = 0.0;
	if (t_3 <= -1e+25)
		tmp = y / (((x + 1.0) * t_2) / z);
	elseif ((t_3 <= 2e-27) || ~((t_3 <= 1e+253)))
		tmp = ((y / (t * (x + 1.0))) + (x / (x + 1.0))) - (x / ((x + 1.0) * (z * t)));
	else
		tmp = (x + (1.0 / (t_2 / t_1))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x + N[(t$95$1 / t$95$2), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -1e+25], N[(y / N[(N[(N[(x + 1.0), $MachinePrecision] * t$95$2), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[t$95$3, 2e-27], N[Not[LessEqual[t$95$3, 1e+253]], $MachinePrecision]], N[(N[(N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(x / N[(N[(x + 1.0), $MachinePrecision] * N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(1.0 / N[(t$95$2 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -1.00000000000000009e25

   1. Initial program 82.0%

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

      1. *-commutative 82.0%

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

   3. Simplified 82.0%

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

   4. Taylor expanded in y around inf 82.1%

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

      1. associate-/l* 97.2%

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

      2. *-commutative 97.2%

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

      3. +-commutative 97.2%

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

   6. Simplified 97.2%

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

   if -1.00000000000000009e25 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 2.0000000000000001e-27 or 9.9999999999999994e252 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 65.9%

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

      1. *-commutative 65.9%

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

   3. Simplified 65.9%

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

   4. Taylor expanded in t around inf 92.1%

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

   if 2.0000000000000001e-27 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 9.9999999999999994e252

   1. Initial program 99.9%

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

      1. *-commutative 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

      2. *-commutative 99.9%

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

      3. *-commutative 99.9%

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

   7. Simplified 99.9%

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

4. Final simplification 97.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -1 \cdot 10^{+25}:\\ \;\;\;\;\frac{y}{\frac{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 2 \cdot 10^{-27} \lor \neg \left(\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+253}\right):\\ \;\;\;\;\left(\frac{y}{t \cdot \left(x + 1\right)} + \frac{x}{x + 1}\right) - \frac{x}{\left(x + 1\right) \cdot \left(z \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{1}{\frac{z \cdot t - x}{y \cdot z - x}}}{x + 1}\\ \end{array} \]

Alternative 2: 96.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ t_2 := \frac{x + \frac{y \cdot z - x}{t_1}}{x + 1}\\ \mathbf{if}\;t_2 \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{\frac{\left(x + 1\right) \cdot t_1}{z}}\\ \mathbf{elif}\;t_2 \leq 10^{+253}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)) (t_2 (/ (+ x (/ (- (* y z) x) t_1)) (+ x 1.0))))
   (if (<= t_2 -5e+17)
     (/ y (/ (* (+ x 1.0) t_1) z))
     (if (<= t_2 1e+253) t_2 (/ (+ x (/ y t)) (+ x 1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+17) {
		tmp = y / (((x + 1.0) * t_1) / z);
	} else if (t_2 <= 1e+253) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (z * t) - x
    t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0d0)
    if (t_2 <= (-5d+17)) then
        tmp = y / (((x + 1.0d0) * t_1) / z)
    else if (t_2 <= 1d+253) then
        tmp = t_2
    else
        tmp = (x + (y / t)) / (x + 1.0d0)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	double t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	double tmp;
	if (t_2 <= -5e+17) {
		tmp = y / (((x + 1.0) * t_1) / z);
	} else if (t_2 <= 1e+253) {
		tmp = t_2;
	} else {
		tmp = (x + (y / t)) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z * t) - x
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0)
	tmp = 0
	if t_2 <= -5e+17:
		tmp = y / (((x + 1.0) * t_1) / z)
	elif t_2 <= 1e+253:
		tmp = t_2
	else:
		tmp = (x + (y / t)) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	t_2 = Float64(Float64(x + Float64(Float64(Float64(y * z) - x) / t_1)) / Float64(x + 1.0))
	tmp = 0.0
	if (t_2 <= -5e+17)
		tmp = Float64(y / Float64(Float64(Float64(x + 1.0) * t_1) / z));
	elseif (t_2 <= 1e+253)
		tmp = t_2;
	else
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z * t) - x;
	t_2 = (x + (((y * z) - x) / t_1)) / (x + 1.0);
	tmp = 0.0;
	if (t_2 <= -5e+17)
		tmp = y / (((x + 1.0) * t_1) / z);
	elseif (t_2 <= 1e+253)
		tmp = t_2;
	else
		tmp = (x + (y / t)) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z * t), $MachinePrecision] - x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x + N[(N[(N[(y * z), $MachinePrecision] - x), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+17], N[(y / N[(N[(N[(x + 1.0), $MachinePrecision] * t$95$1), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+253], t$95$2, N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < -5e17

   1. Initial program 82.4%

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

      1. *-commutative 82.4%

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

   3. Simplified 82.4%

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

   4. Taylor expanded in y around inf 82.5%

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

      1. associate-/l* 97.3%

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

      2. *-commutative 97.3%

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

      3. +-commutative 97.3%

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

   6. Simplified 97.3%

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

   if -5e17 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1)) < 9.9999999999999994e252

   1. Initial program 97.4%

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

   if 9.9999999999999994e252 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x 1))

   1. Initial program 16.9%

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

      1. *-commutative 16.9%

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

   3. Simplified 16.9%

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

   4. Taylor expanded in z around inf 75.2%

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

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq -5 \cdot 10^{+17}:\\ \;\;\;\;\frac{y}{\frac{\left(x + 1\right) \cdot \left(z \cdot t - x\right)}{z}}\\ \mathbf{elif}\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1} \leq 10^{+253}:\\ \;\;\;\;\frac{x + \frac{y \cdot z - x}{z \cdot t - x}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \end{array} \]

Alternative 3: 96.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot t - x\\ \frac{y \cdot \frac{z}{t_1} + \left(x - \frac{x}{t_1}\right)}{x + 1} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* z t) x)))
   (/ (+ (* y (/ z t_1)) (- x (/ x t_1))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	return ((y * (z / t_1)) + (x - (x / t_1))) / (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
    real(8) :: t_1
    t_1 = (z * t) - x
    code = ((y * (z / t_1)) + (x - (x / t_1))) / (x + 1.0d0)
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (z * t) - x;
	return ((y * (z / t_1)) + (x - (x / t_1))) / (x + 1.0);
}

Python

def code(x, y, z, t):
	t_1 = (z * t) - x
	return ((y * (z / t_1)) + (x - (x / t_1))) / (x + 1.0)

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z * t) - x)
	return Float64(Float64(Float64(y * Float64(z / t_1)) + Float64(x - Float64(x / t_1))) / Float64(x + 1.0))
end

MATLAB

function tmp = code(x, y, z, t)
	t_1 = (z * t) - x;
	tmp = ((y * (z / t_1)) + (x - (x / t_1))) / (x + 1.0);
end

Wolfram

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

Derivation

1. Initial program 88.9%

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

   1. *-commutative 88.9%

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

3. Simplified 88.9%

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

   1. +-commutative 88.9%

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

   2. div-sub 88.9%

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

   3. associate-+l- 88.9%

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

   4. *-un-lft-identity 88.9%

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

   5. times-frac 94.5%

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

   6. fma-neg 94.5%

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

5. Applied egg-rr 94.5%

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

   1. fma-udef 94.5%

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

   2. /-rgt-identity 94.5%

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

7. Applied egg-rr 94.5%

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

8. Final simplification 94.5%

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

Alternative 4: 79.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-175} \lor \neg \left(z \leq 4.3 \cdot 10^{-128}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{z \cdot t - x}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6e-175) (not (<= z 4.3e-128)))
   (/ (+ x (/ y t)) (+ x 1.0))
   (/ (- x (/ x (- (* z t) x))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6e-175) || !(z <= 4.3e-128)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x - (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 <= (-6d-175)) .or. (.not. (z <= 4.3d-128))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = (x - (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 <= -6e-175) || !(z <= 4.3e-128)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = (x - (x / ((z * t) - x))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -6e-175) or not (z <= 4.3e-128):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = (x - (x / ((z * t) - x))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6e-175) || !(z <= 4.3e-128))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = Float64(Float64(x - Float64(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 <= -6e-175) || ~((z <= 4.3e-128)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = (x - (x / ((z * t) - x))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6e-175], N[Not[LessEqual[z, 4.3e-128]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], N[(N[(x - N[(x / 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 < -6e-175 or 4.29999999999999994e-128 < z

   1. Initial program 84.1%

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

      1. *-commutative 84.1%

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

   3. Simplified 84.1%

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

   4. Taylor expanded in z around inf 81.7%

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

   if -6e-175 < z < 4.29999999999999994e-128

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in y around 0 79.0%

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

      1. +-commutative 79.0%

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

   6. Simplified 79.0%

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

4. Final simplification 80.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-175} \lor \neg \left(z \leq 4.3 \cdot 10^{-128}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x - \frac{x}{z \cdot t - x}}{x + 1}\\ \end{array} \]

Alternative 5: 75.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-175} \lor \neg \left(z \leq 3.15 \cdot 10^{-223}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.5e-175) (not (<= z 3.15e-223)))
   (/ (+ x (/ y t)) (+ x 1.0))
   1.0))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.5e-175) || !(z <= 3.15e-223)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 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 <= (-2.5d-175)) .or. (.not. (z <= 3.15d-223))) then
        tmp = (x + (y / t)) / (x + 1.0d0)
    else
        tmp = 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 <= -2.5e-175) || !(z <= 3.15e-223)) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.5e-175) or not (z <= 3.15e-223):
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.5e-175) || !(z <= 3.15e-223))
		tmp = Float64(Float64(x + Float64(y / t)) / Float64(x + 1.0));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.5e-175) || ~((z <= 3.15e-223)))
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.5e-175], N[Not[LessEqual[z, 3.15e-223]], $MachinePrecision]], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]

Derivation

1. Split input into 2 regimes

2. if z < -2.5e-175 or 3.14999999999999993e-223 < z

   1. Initial program 85.7%

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

      1. *-commutative 85.7%

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

   3. Simplified 85.7%

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

   4. Taylor expanded in z around inf 80.2%

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

   if -2.5e-175 < z < 3.14999999999999993e-223

   1. Initial program 99.8%

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

      1. *-commutative 99.8%

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

   3. Simplified 99.8%

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

      1. +-commutative 99.8%

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

      2. div-sub 99.8%

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

      3. associate-+l- 99.8%

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

      4. *-un-lft-identity 99.8%

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

      5. times-frac 99.8%

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

      6. fma-neg 99.8%

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

   5. Applied egg-rr 99.8%

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

   6. Taylor expanded in z around 0 69.8%

      \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{-175} \lor \neg \left(z \leq 3.15 \cdot 10^{-223}\right):\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 6: 68.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-42}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{z}{x} \cdot \frac{y}{x}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.25e-72)
   (/ x (+ x 1.0))
   (if (<= x 5e-42) (/ y t) (- 1.0 (* (/ z x) (/ y x))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.25e-72) {
		tmp = x / (x + 1.0);
	} else if (x <= 5e-42) {
		tmp = y / t;
	} else {
		tmp = 1.0 - ((z / x) * (y / x));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-3.25d-72)) then
        tmp = x / (x + 1.0d0)
    else if (x <= 5d-42) then
        tmp = y / t
    else
        tmp = 1.0d0 - ((z / x) * (y / x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.25e-72) {
		tmp = x / (x + 1.0);
	} else if (x <= 5e-42) {
		tmp = y / t;
	} else {
		tmp = 1.0 - ((z / x) * (y / x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.25e-72:
		tmp = x / (x + 1.0)
	elif x <= 5e-42:
		tmp = y / t
	else:
		tmp = 1.0 - ((z / x) * (y / x))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.25e-72)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 5e-42)
		tmp = Float64(y / t);
	else
		tmp = Float64(1.0 - Float64(Float64(z / x) * Float64(y / x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.25e-72)
		tmp = x / (x + 1.0);
	elseif (x <= 5e-42)
		tmp = y / t;
	else
		tmp = 1.0 - ((z / x) * (y / x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.25e-72], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-42], N[(y / t), $MachinePrecision], N[(1.0 - N[(N[(z / x), $MachinePrecision] * N[(y / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.2499999999999998e-72

   1. Initial program 90.4%

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

      1. *-commutative 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in t around inf 82.1%

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

      1. +-commutative 82.1%

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

   6. Simplified 82.1%

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

   if -3.2499999999999998e-72 < x < 5.00000000000000003e-42

   1. Initial program 88.5%

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

      1. *-commutative 88.5%

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

   3. Simplified 88.5%

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

      1. +-commutative 88.5%

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

      2. div-sub 88.5%

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

      3. associate-+l- 88.5%

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

      4. *-un-lft-identity 88.5%

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

      5. times-frac 91.9%

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

      6. fma-neg 91.9%

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

   5. Applied egg-rr 91.9%

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

   6. Taylor expanded in x around 0 56.7%

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

   if 5.00000000000000003e-42 < x

   1. Initial program 87.6%

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

      1. *-commutative 87.6%

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

   3. Simplified 87.6%

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

   4. Taylor expanded in x around -inf 60.9%

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

      1. mul-1-neg 60.9%

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

      2. unsub-neg 60.9%

         \[\leadsto \color{blue}{1 - \frac{y \cdot z - t \cdot z}{{x}^{2}}} \]

      3. distribute-rgt-out-- 60.9%

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

      4. unpow2 60.9%

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

   6. Simplified 60.9%

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

   7. Taylor expanded in y around inf 74.0%

      \[\leadsto 1 - \color{blue}{\frac{y \cdot z}{{x}^{2}}} \]
   8. Step-by-step derivation

      1. *-commutative 74.0%

         \[\leadsto 1 - \frac{\color{blue}{z \cdot y}}{{x}^{2}} \]

      2. unpow2 74.0%

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

   9. Simplified 74.0%

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

       1. times-frac 78.7%

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

   11. Applied egg-rr 78.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.25 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-42}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{z}{x} \cdot \frac{y}{x}\\ \end{array} \]

Alternative 7: 67.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-35}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-73}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4e-35)
   1.0
   (if (<= x -2.65e-73) (- x (* x x)) (if (<= x 4.6e-41) (/ y t) 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4e-35) {
		tmp = 1.0;
	} else if (x <= -2.65e-73) {
		tmp = x - (x * x);
	} else if (x <= 4.6e-41) {
		tmp = y / t;
	} else {
		tmp = 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 <= (-4d-35)) then
        tmp = 1.0d0
    else if (x <= (-2.65d-73)) then
        tmp = x - (x * x)
    else if (x <= 4.6d-41) then
        tmp = y / t
    else
        tmp = 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 <= -4e-35) {
		tmp = 1.0;
	} else if (x <= -2.65e-73) {
		tmp = x - (x * x);
	} else if (x <= 4.6e-41) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4e-35:
		tmp = 1.0
	elif x <= -2.65e-73:
		tmp = x - (x * x)
	elif x <= 4.6e-41:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4e-35)
		tmp = 1.0;
	elseif (x <= -2.65e-73)
		tmp = Float64(x - Float64(x * x));
	elseif (x <= 4.6e-41)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4e-35)
		tmp = 1.0;
	elseif (x <= -2.65e-73)
		tmp = x - (x * x);
	elseif (x <= 4.6e-41)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4e-35], 1.0, If[LessEqual[x, -2.65e-73], N[(x - N[(x * x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 4.6e-41], N[(y / t), $MachinePrecision], 1.0]]]

Derivation

1. Split input into 3 regimes

2. if x < -4.00000000000000003e-35 or 4.6000000000000002e-41 < x

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

   3. Simplified 88.7%

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

      1. +-commutative 88.7%

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

      2. div-sub 88.7%

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

      3. associate-+l- 88.7%

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

      4. *-un-lft-identity 88.7%

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

      5. times-frac 96.4%

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

      6. fma-neg 96.4%

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

   5. Applied egg-rr 96.4%

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

   6. Taylor expanded in z around 0 80.6%

      \[\leadsto \color{blue}{1} \]

   if -4.00000000000000003e-35 < x < -2.64999999999999986e-73

   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 inf 84.1%

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

      1. +-commutative 84.1%

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

   6. Simplified 84.1%

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

   7. Taylor expanded in x around 0 84.1%

      \[\leadsto \color{blue}{-1 \cdot {x}^{2} + x} \]
   8. Step-by-step derivation

      1. +-commutative 84.1%

         \[\leadsto \color{blue}{x + -1 \cdot {x}^{2}} \]

      2. mul-1-neg 84.1%

         \[\leadsto x + \color{blue}{\left(-{x}^{2}\right)} \]

      3. unsub-neg 84.1%

         \[\leadsto \color{blue}{x - {x}^{2}} \]

      4. unpow2 84.1%

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

   9. Simplified 84.1%

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

   if -2.64999999999999986e-73 < x < 4.6000000000000002e-41

   1. Initial program 88.5%

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

      1. *-commutative 88.5%

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

   3. Simplified 88.5%

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

      1. +-commutative 88.5%

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

      2. div-sub 88.5%

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

      3. associate-+l- 88.5%

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

      4. *-un-lft-identity 88.5%

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

      5. times-frac 91.9%

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

      6. fma-neg 91.9%

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

   5. Applied egg-rr 91.9%

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

   6. Taylor expanded in x around 0 56.7%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -4 \cdot 10^{-35}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq -2.65 \cdot 10^{-73}:\\ \;\;\;\;x - x \cdot x\\ \mathbf{elif}\;x \leq 4.6 \cdot 10^{-41}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 8: 67.4% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-36}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.7e-36) 1.0 (if (<= x 5.8e-36) (/ y t) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.7e-36) {
		tmp = 1.0;
	} else if (x <= 5.8e-36) {
		tmp = y / t;
	} else {
		tmp = 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 <= (-2.7d-36)) then
        tmp = 1.0d0
    else if (x <= 5.8d-36) then
        tmp = y / t
    else
        tmp = 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 <= -2.7e-36) {
		tmp = 1.0;
	} else if (x <= 5.8e-36) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.7e-36:
		tmp = 1.0
	elif x <= 5.8e-36:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.7e-36)
		tmp = 1.0;
	elseif (x <= 5.8e-36)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.7e-36)
		tmp = 1.0;
	elseif (x <= 5.8e-36)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.7e-36], 1.0, If[LessEqual[x, 5.8e-36], N[(y / t), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -2.70000000000000007e-36 or 5.80000000000000026e-36 < x

   1. Initial program 88.7%

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

      1. *-commutative 88.7%

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

   3. Simplified 88.7%

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

      1. +-commutative 88.7%

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

      2. div-sub 88.7%

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

      3. associate-+l- 88.7%

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

      4. *-un-lft-identity 88.7%

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

      5. times-frac 96.4%

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

      6. fma-neg 96.4%

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

   5. Applied egg-rr 96.4%

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

   6. Taylor expanded in z around 0 80.6%

      \[\leadsto \color{blue}{1} \]

   if -2.70000000000000007e-36 < x < 5.80000000000000026e-36

   1. Initial program 89.0%

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

      1. *-commutative 89.0%

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

   3. Simplified 89.0%

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

      1. +-commutative 89.0%

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

      2. div-sub 89.0%

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

      3. associate-+l- 89.0%

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

      4. *-un-lft-identity 89.0%

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

      5. times-frac 92.3%

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

      6. fma-neg 92.3%

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

   5. Applied egg-rr 92.3%

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

   6. Taylor expanded in x around 0 54.0%

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

4. Final simplification 68.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.7 \cdot 10^{-36}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 5.8 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 9: 67.7% accurate, 2.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.65e-72) (/ x (+ x 1.0)) (if (<= x 5e-36) (/ y t) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.65e-72) {
		tmp = x / (x + 1.0);
	} else if (x <= 5e-36) {
		tmp = y / t;
	} else {
		tmp = 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 <= (-1.65d-72)) then
        tmp = x / (x + 1.0d0)
    else if (x <= 5d-36) then
        tmp = y / t
    else
        tmp = 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 <= -1.65e-72) {
		tmp = x / (x + 1.0);
	} else if (x <= 5e-36) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.65e-72:
		tmp = x / (x + 1.0)
	elif x <= 5e-36:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.65e-72)
		tmp = Float64(x / Float64(x + 1.0));
	elseif (x <= 5e-36)
		tmp = Float64(y / t);
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -1.65e-72)
		tmp = x / (x + 1.0);
	elseif (x <= 5e-36)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.65e-72], N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 5e-36], N[(y / t), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65e-72

   1. Initial program 90.4%

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

      1. *-commutative 90.4%

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

   3. Simplified 90.4%

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

   4. Taylor expanded in t around inf 82.1%

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

      1. +-commutative 82.1%

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

   6. Simplified 82.1%

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

   if -1.65e-72 < x < 5.00000000000000004e-36

   1. Initial program 88.5%

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

      1. *-commutative 88.5%

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

   3. Simplified 88.5%

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

      1. +-commutative 88.5%

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

      2. div-sub 88.5%

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

      3. associate-+l- 88.5%

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

      4. *-un-lft-identity 88.5%

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

      5. times-frac 91.9%

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

      6. fma-neg 91.9%

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

   5. Applied egg-rr 91.9%

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

   6. Taylor expanded in x around 0 56.7%

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

   if 5.00000000000000004e-36 < x

   1. Initial program 87.6%

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

      1. *-commutative 87.6%

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

   3. Simplified 87.6%

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

      1. +-commutative 87.6%

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

      2. div-sub 87.6%

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

      3. associate-+l- 87.6%

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

      4. *-un-lft-identity 87.6%

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

      5. times-frac 95.4%

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

      6. fma-neg 95.4%

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

   5. Applied egg-rr 95.4%

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

   6. Taylor expanded in z around 0 77.8%

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

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{-72}:\\ \;\;\;\;\frac{x}{x + 1}\\ \mathbf{elif}\;x \leq 5 \cdot 10^{-36}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 10: 52.3% accurate, 17.0× speedup

Math

\[\begin{array}{l} \\ 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 1.0)

C

double code(double x, double y, double z, double t) {
	return 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 = 1.0d0
end function

Java

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

Python

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

Julia

function code(x, y, z, t)
	return 1.0
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := 1.0

Derivation

1. Initial program 88.9%

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

   1. *-commutative 88.9%

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

3. Simplified 88.9%

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

   1. +-commutative 88.9%

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

   2. div-sub 88.9%

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

   3. associate-+l- 88.9%

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

   4. *-un-lft-identity 88.9%

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

   5. times-frac 94.5%

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

   6. fma-neg 94.5%

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

5. Applied egg-rr 94.5%

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

6. Taylor expanded in z around 0 51.2%

   \[\leadsto \color{blue}{1} \]

7. Final simplification 51.2%

   \[\leadsto 1 \]

Developer target: 99.5% 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 2023252 
(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)))