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

Percentage Accurate: 89.4% → 94.0%

Time: 15.1s

Alternatives: 11

Speedup: 0.4×

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: 89.4% 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: 94.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{if}\;t\_1 \leq 10^{+209}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ x (/ (- x (* y z)) (- x (* z t)))) (+ x 1.0))))
   (if (<= t_1 1e+209) t_1 (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_1 <= 1e+209) {
		tmp = t_1;
	} else {
		tmp = (x / (x + 1.0)) + (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) :: tmp
    t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0d0)
    if (t_1 <= 1d+209) then
        tmp = t_1
    else
        tmp = (x / (x + 1.0d0)) + (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 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	double tmp;
	if (t_1 <= 1e+209) {
		tmp = t_1;
	} else {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0)
	tmp = 0
	if t_1 <= 1e+209:
		tmp = t_1
	else:
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x + Float64(Float64(x - Float64(y * z)) / Float64(x - Float64(z * t)))) / Float64(x + 1.0))
	tmp = 0.0
	if (t_1 <= 1e+209)
		tmp = t_1;
	else
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * Float64(x + 1.0))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x + ((x - (y * z)) / (x - (z * t)))) / (x + 1.0);
	tmp = 0.0;
	if (t_1 <= 1e+209)
		tmp = t_1;
	else
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x + N[(N[(x - N[(y * z), $MachinePrecision]), $MachinePrecision] / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+209], t$95$1, N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64))) < 1.0000000000000001e209

   1. Initial program 98.2%

      \[\frac{x + \frac{y \cdot z - x}{t \cdot z - x}}{x + 1} \]
   2. Add Preprocessing

   if 1.0000000000000001e209 < (/.f64 (+.f64 x (/.f64 (-.f64 (*.f64 y z) x) (-.f64 (*.f64 t z) x))) (+.f64 x #s(literal 1 binary64)))

   1. Initial program 24.9%

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

      1. *-commutative 24.9%

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

   3. Simplified 24.9%

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

   5. Taylor expanded in t around inf 93.6%

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

      1. +-commutative 93.6%

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

      2. +-commutative 93.6%

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

      3. +-commutative 93.6%

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

   7. Simplified 93.6%

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

   8. Taylor expanded in z around inf 93.6%

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

      1. +-commutative 93.6%

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

      2. +-commutative 93.6%

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

   10. Simplified 93.6%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1} \leq 10^{+209}:\\ \;\;\;\;\frac{x + \frac{x - y \cdot z}{x - z \cdot t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 77.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\left(x + 1\right) - y \cdot \frac{z}{x}}{x + 1}\\ \mathbf{if}\;x \leq -1.25 \cdot 10^{-130}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{\frac{x}{z} - y}{t} - x}{-1 - x}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- (+ x 1.0) (* y (/ z x))) (+ x 1.0))))
   (if (<= x -1.25e-130)
     t_1
     (if (<= x 1.85e-94)
       (/ (- (/ (- (/ x z) y) t) x) (- -1.0 x))
       (if (<= x 7.8e-41)
         t_1
         (if (<= x 1.25e+40)
           (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0))))
           1.0))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x + 1.0) - (y * (z / x))) / (x + 1.0);
	double tmp;
	if (x <= -1.25e-130) {
		tmp = t_1;
	} else if (x <= 1.85e-94) {
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x);
	} else if (x <= 7.8e-41) {
		tmp = t_1;
	} else if (x <= 1.25e+40) {
		tmp = (x / (x + 1.0)) + (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) :: t_1
    real(8) :: tmp
    t_1 = ((x + 1.0d0) - (y * (z / x))) / (x + 1.0d0)
    if (x <= (-1.25d-130)) then
        tmp = t_1
    else if (x <= 1.85d-94) then
        tmp = ((((x / z) - y) / t) - x) / ((-1.0d0) - x)
    else if (x <= 7.8d-41) then
        tmp = t_1
    else if (x <= 1.25d+40) then
        tmp = (x / (x + 1.0d0)) + (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 t_1 = ((x + 1.0) - (y * (z / x))) / (x + 1.0);
	double tmp;
	if (x <= -1.25e-130) {
		tmp = t_1;
	} else if (x <= 1.85e-94) {
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x);
	} else if (x <= 7.8e-41) {
		tmp = t_1;
	} else if (x <= 1.25e+40) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((x + 1.0) - (y * (z / x))) / (x + 1.0)
	tmp = 0
	if x <= -1.25e-130:
		tmp = t_1
	elif x <= 1.85e-94:
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x)
	elif x <= 7.8e-41:
		tmp = t_1
	elif x <= 1.25e+40:
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x + 1.0) - Float64(y * Float64(z / x))) / Float64(x + 1.0))
	tmp = 0.0
	if (x <= -1.25e-130)
		tmp = t_1;
	elseif (x <= 1.85e-94)
		tmp = Float64(Float64(Float64(Float64(Float64(x / z) - y) / t) - x) / Float64(-1.0 - x));
	elseif (x <= 7.8e-41)
		tmp = t_1;
	elseif (x <= 1.25e+40)
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * Float64(x + 1.0))));
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((x + 1.0) - (y * (z / x))) / (x + 1.0);
	tmp = 0.0;
	if (x <= -1.25e-130)
		tmp = t_1;
	elseif (x <= 1.85e-94)
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x);
	elseif (x <= 7.8e-41)
		tmp = t_1;
	elseif (x <= 1.25e+40)
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x + 1.0), $MachinePrecision] - N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.25e-130], t$95$1, If[LessEqual[x, 1.85e-94], N[(N[(N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 7.8e-41], t$95$1, If[LessEqual[x, 1.25e+40], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]]]]

Derivation

1. Split input into 4 regimes

2. if x < -1.2499999999999999e-130 or 1.8499999999999999e-94 < x < 7.79999999999999982e-41

   1. Initial program 90.3%

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

      1. *-commutative 90.3%

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

   3. Simplified 90.3%

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

   5. Taylor expanded in t around 0 78.7%

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

      1. associate-+r+ 78.7%

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

      2. mul-1-neg 78.7%

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

      3. unsub-neg 78.7%

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

      4. +-commutative 78.7%

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

      5. associate-/l* 85.5%

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

      6. +-commutative 85.5%

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

   7. Simplified 85.5%

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

   if -1.2499999999999999e-130 < x < 1.8499999999999999e-94

   1. Initial program 89.2%

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

      1. *-commutative 89.2%

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

   3. Simplified 89.2%

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

   5. Taylor expanded in t around -inf 87.4%

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

      1. mul-1-neg 87.4%

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

      2. unsub-neg 87.4%

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

      3. cancel-sign-sub-inv 87.4%

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

      4. metadata-eval 87.4%

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

      5. *-lft-identity 87.4%

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

      6. +-commutative 87.4%

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

      7. mul-1-neg 87.4%

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

      8. unsub-neg 87.4%

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

   7. Simplified 87.4%

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

   if 7.79999999999999982e-41 < x < 1.25000000000000001e40

   1. Initial program 93.1%

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

      1. *-commutative 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in t around inf 62.7%

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

      1. +-commutative 62.7%

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

      2. +-commutative 62.7%

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

      3. +-commutative 62.7%

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

   7. Simplified 62.7%

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

   8. Taylor expanded in z around inf 64.4%

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

      1. +-commutative 64.4%

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

      2. +-commutative 64.4%

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

   10. Simplified 64.4%

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

   if 1.25000000000000001e40 < x

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

   3. Simplified 88.0%

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

      1. clear-num 88.0%

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

      2. inv-pow 88.0%

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

      3. fma-neg 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. unpow-1 88.0%

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

      2. *-commutative 88.0%

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

      3. fma-neg 88.0%

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

      4. *-commutative 88.0%

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

   8. Simplified 88.0%

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

      1. div-inv 88.0%

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

      2. fma-neg 88.0%

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

   10. Applied egg-rr 88.0%

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

   11. Taylor expanded in z around inf 88.0%

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

   12. Taylor expanded in x around inf 100.0%

       \[\leadsto \color{blue}{1} \]
3. Recombined 4 regimes into one program.

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.25 \cdot 10^{-130}:\\ \;\;\;\;\frac{\left(x + 1\right) - y \cdot \frac{z}{x}}{x + 1}\\ \mathbf{elif}\;x \leq 1.85 \cdot 10^{-94}:\\ \;\;\;\;\frac{\frac{\frac{x}{z} - y}{t} - x}{-1 - x}\\ \mathbf{elif}\;x \leq 7.8 \cdot 10^{-41}:\\ \;\;\;\;\frac{\left(x + 1\right) - y \cdot \frac{z}{x}}{x + 1}\\ \mathbf{elif}\;x \leq 1.25 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 80.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-58} \lor \neg \left(z \leq 2.8 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -5.8e-58) (not (<= z 2.8e-89)))
   (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0))))
   (/ (+ x (/ x (- x (* z t)))) (+ x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -5.8e-58) || !(z <= 2.8e-89)) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else {
		tmp = (x + (x / (x - (z * 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) :: tmp
    if ((z <= (-5.8d-58)) .or. (.not. (z <= 2.8d-89))) then
        tmp = (x / (x + 1.0d0)) + (y / (t * (x + 1.0d0)))
    else
        tmp = (x + (x / (x - (z * t)))) / (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 <= -5.8e-58) || !(z <= 2.8e-89)) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -5.8e-58) or not (z <= 2.8e-89):
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)))
	else:
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -5.8e-58) || !(z <= 2.8e-89))
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(t * Float64(x + 1.0))));
	else
		tmp = Float64(Float64(x + Float64(x / Float64(x - Float64(z * t)))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -5.8e-58) || ~((z <= 2.8e-89)))
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	else
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -5.8e-58], N[Not[LessEqual[z, 2.8e-89]], $MachinePrecision]], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(x / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.7999999999999998e-58 or 2.7999999999999999e-89 < z

   1. Initial program 83.6%

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

      1. *-commutative 83.6%

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

   3. Simplified 83.6%

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

   5. Taylor expanded in t around inf 82.1%

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

      1. +-commutative 82.1%

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

      2. +-commutative 82.1%

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

      3. +-commutative 82.1%

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

   7. Simplified 82.1%

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

   8. Taylor expanded in z around inf 83.6%

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

      1. +-commutative 83.6%

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

      2. +-commutative 83.6%

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

   10. Simplified 83.6%

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

   if -5.7999999999999998e-58 < z < 2.7999999999999999e-89

   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. Add Preprocessing

   5. Taylor expanded in y around 0 82.3%

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

      1. +-commutative 82.3%

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

   7. Simplified 82.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-58} \lor \neg \left(z \leq 2.8 \cdot 10^{-89}\right):\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 80.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3:\\ \;\;\;\;\frac{y \cdot \frac{z}{x} - x}{-1 - x}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{\frac{x}{z} - y}{t} - x}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.0)
   (/ (- (* y (/ z x)) x) (- -1.0 x))
   (if (<= x 1.15e-49)
     (/ (- (/ (- (/ x z) y) t) x) (- -1.0 x))
     (/ (+ x (/ x (- x (* z t)))) (+ x 1.0)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.0) {
		tmp = ((y * (z / x)) - x) / (-1.0 - x);
	} else if (x <= 1.15e-49) {
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x);
	} else {
		tmp = (x + (x / (x - (z * 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) :: tmp
    if (x <= (-3.0d0)) then
        tmp = ((y * (z / x)) - x) / ((-1.0d0) - x)
    else if (x <= 1.15d-49) then
        tmp = ((((x / z) - y) / t) - x) / ((-1.0d0) - x)
    else
        tmp = (x + (x / (x - (z * t)))) / (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 <= -3.0) {
		tmp = ((y * (z / x)) - x) / (-1.0 - x);
	} else if (x <= 1.15e-49) {
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x);
	} else {
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.0:
		tmp = ((y * (z / x)) - x) / (-1.0 - x)
	elif x <= 1.15e-49:
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x)
	else:
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.0)
		tmp = Float64(Float64(Float64(y * Float64(z / x)) - x) / Float64(-1.0 - x));
	elseif (x <= 1.15e-49)
		tmp = Float64(Float64(Float64(Float64(Float64(x / z) - y) / t) - x) / Float64(-1.0 - x));
	else
		tmp = Float64(Float64(x + Float64(x / Float64(x - Float64(z * t)))) / Float64(x + 1.0));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.0)
		tmp = ((y * (z / x)) - x) / (-1.0 - x);
	elseif (x <= 1.15e-49)
		tmp = ((((x / z) - y) / t) - x) / (-1.0 - x);
	else
		tmp = (x + (x / (x - (z * t)))) / (x + 1.0);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.0], N[(N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.15e-49], N[(N[(N[(N[(N[(x / z), $MachinePrecision] - y), $MachinePrecision] / t), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], N[(N[(x + N[(x / N[(x - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

   3. Simplified 87.2%

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

      1. clear-num 87.2%

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

      2. inv-pow 87.2%

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

      3. fma-neg 87.2%

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

   6. Applied egg-rr 87.2%

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

      1. unpow-1 87.2%

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

      2. *-commutative 87.2%

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

      3. fma-neg 87.2%

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

      4. *-commutative 87.2%

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

   8. Simplified 87.2%

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

      1. div-inv 87.2%

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

      2. fma-neg 87.2%

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

   10. Applied egg-rr 87.2%

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

   11. Taylor expanded in z around inf 84.6%

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

   12. Taylor expanded in t around 0 80.4%

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

       1. mul-1-neg 80.4%

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

       2. unsub-neg 80.4%

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

       3. associate-/l* 90.3%

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

       4. +-commutative 90.3%

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

   14. Simplified 90.3%

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

   if -3 < x < 1.15e-49

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in t around -inf 78.6%

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

      1. mul-1-neg 78.6%

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

      2. unsub-neg 78.6%

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

      3. cancel-sign-sub-inv 78.6%

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

      4. metadata-eval 78.6%

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

      5. *-lft-identity 78.6%

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

      6. +-commutative 78.6%

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

      7. mul-1-neg 78.6%

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

      8. unsub-neg 78.6%

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

   7. Simplified 78.6%

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

   if 1.15e-49 < x

   1. Initial program 89.5%

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

      1. *-commutative 89.5%

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

   3. Simplified 89.5%

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

   5. Taylor expanded in y around 0 89.8%

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

      1. +-commutative 89.8%

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

   7. Simplified 89.8%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3:\\ \;\;\;\;\frac{y \cdot \frac{z}{x} - x}{-1 - x}\\ \mathbf{elif}\;x \leq 1.15 \cdot 10^{-49}:\\ \;\;\;\;\frac{\frac{\frac{x}{z} - y}{t} - x}{-1 - x}\\ \mathbf{else}:\\ \;\;\;\;\frac{x + \frac{x}{x - z \cdot t}}{x + 1}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 77.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3400:\\ \;\;\;\;\frac{y \cdot \frac{z}{x} - x}{-1 - x}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3400.0)
   (/ (- (* y (/ z x)) x) (- -1.0 x))
   (if (<= x 1.95e+40) (+ (/ x (+ x 1.0)) (/ y (* t (+ x 1.0)))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3400.0) {
		tmp = ((y * (z / x)) - x) / (-1.0 - x);
	} else if (x <= 1.95e+40) {
		tmp = (x / (x + 1.0)) + (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 (x <= (-3400.0d0)) then
        tmp = ((y * (z / x)) - x) / ((-1.0d0) - x)
    else if (x <= 1.95d+40) then
        tmp = (x / (x + 1.0d0)) + (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 (x <= -3400.0) {
		tmp = ((y * (z / x)) - x) / (-1.0 - x);
	} else if (x <= 1.95e+40) {
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3400.0:
		tmp = ((y * (z / x)) - x) / (-1.0 - x)
	elif x <= 1.95e+40:
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3400.0)
		tmp = Float64(Float64(Float64(y * Float64(z / x)) - x) / Float64(-1.0 - x));
	elseif (x <= 1.95e+40)
		tmp = Float64(Float64(x / Float64(x + 1.0)) + Float64(y / Float64(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 (x <= -3400.0)
		tmp = ((y * (z / x)) - x) / (-1.0 - x);
	elseif (x <= 1.95e+40)
		tmp = (x / (x + 1.0)) + (y / (t * (x + 1.0)));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3400.0], N[(N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.95e+40], N[(N[(x / N[(x + 1.0), $MachinePrecision]), $MachinePrecision] + N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -3400

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

   3. Simplified 87.2%

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

      1. clear-num 87.2%

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

      2. inv-pow 87.2%

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

      3. fma-neg 87.2%

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

   6. Applied egg-rr 87.2%

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

      1. unpow-1 87.2%

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

      2. *-commutative 87.2%

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

      3. fma-neg 87.2%

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

      4. *-commutative 87.2%

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

   8. Simplified 87.2%

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

      1. div-inv 87.2%

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

      2. fma-neg 87.2%

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

   10. Applied egg-rr 87.2%

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

   11. Taylor expanded in z around inf 84.6%

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

   12. Taylor expanded in t around 0 80.4%

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

       1. mul-1-neg 80.4%

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

       2. unsub-neg 80.4%

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

       3. associate-/l* 90.3%

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

       4. +-commutative 90.3%

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

   14. Simplified 90.3%

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

   if -3400 < x < 1.95e40

   1. Initial program 91.6%

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

      1. *-commutative 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in t around inf 75.0%

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

      1. +-commutative 75.0%

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

      2. +-commutative 75.0%

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

      3. +-commutative 75.0%

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

   7. Simplified 75.0%

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

   8. Taylor expanded in z around inf 67.1%

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

      1. +-commutative 67.1%

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

      2. +-commutative 67.1%

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

   10. Simplified 67.1%

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

   if 1.95e40 < x

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

   3. Simplified 88.0%

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

      1. clear-num 88.0%

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

      2. inv-pow 88.0%

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

      3. fma-neg 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. unpow-1 88.0%

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

      2. *-commutative 88.0%

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

      3. fma-neg 88.0%

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

      4. *-commutative 88.0%

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

   8. Simplified 88.0%

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

      1. div-inv 88.0%

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

      2. fma-neg 88.0%

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

   10. Applied egg-rr 88.0%

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

   11. Taylor expanded in z around inf 88.0%

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

   12. Taylor expanded in x around inf 100.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3400:\\ \;\;\;\;\frac{y \cdot \frac{z}{x} - x}{-1 - x}\\ \mathbf{elif}\;x \leq 1.95 \cdot 10^{+40}:\\ \;\;\;\;\frac{x}{x + 1} + \frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 77.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -40000:\\ \;\;\;\;\frac{y \cdot \frac{z}{x} - x}{-1 - x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+41}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -40000.0)
   (/ (- (* y (/ z x)) x) (- -1.0 x))
   (if (<= x 1.55e+41) (/ (+ x (/ y t)) (+ x 1.0)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -40000.0) {
		tmp = ((y * (z / x)) - x) / (-1.0 - x);
	} else if (x <= 1.55e+41) {
		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 (x <= (-40000.0d0)) then
        tmp = ((y * (z / x)) - x) / ((-1.0d0) - x)
    else if (x <= 1.55d+41) 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 (x <= -40000.0) {
		tmp = ((y * (z / x)) - x) / (-1.0 - x);
	} else if (x <= 1.55e+41) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -40000.0:
		tmp = ((y * (z / x)) - x) / (-1.0 - x)
	elif x <= 1.55e+41:
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -40000.0)
		tmp = Float64(Float64(Float64(y * Float64(z / x)) - x) / Float64(-1.0 - x));
	elseif (x <= 1.55e+41)
		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 (x <= -40000.0)
		tmp = ((y * (z / x)) - x) / (-1.0 - x);
	elseif (x <= 1.55e+41)
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -40000.0], N[(N[(N[(y * N[(z / x), $MachinePrecision]), $MachinePrecision] - x), $MachinePrecision] / N[(-1.0 - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1.55e+41], N[(N[(x + N[(y / t), $MachinePrecision]), $MachinePrecision] / N[(x + 1.0), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 3 regimes

2. if x < -4e4

   1. Initial program 87.2%

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

      1. *-commutative 87.2%

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

   3. Simplified 87.2%

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

      1. clear-num 87.2%

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

      2. inv-pow 87.2%

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

      3. fma-neg 87.2%

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

   6. Applied egg-rr 87.2%

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

      1. unpow-1 87.2%

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

      2. *-commutative 87.2%

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

      3. fma-neg 87.2%

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

      4. *-commutative 87.2%

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

   8. Simplified 87.2%

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

      1. div-inv 87.2%

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

      2. fma-neg 87.2%

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

   10. Applied egg-rr 87.2%

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

   11. Taylor expanded in z around inf 84.6%

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

   12. Taylor expanded in t around 0 80.4%

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

       1. mul-1-neg 80.4%

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

       2. unsub-neg 80.4%

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

       3. associate-/l* 90.3%

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

       4. +-commutative 90.3%

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

   14. Simplified 90.3%

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

   if -4e4 < x < 1.55e41

   1. Initial program 91.6%

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

      1. *-commutative 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in z around inf 67.1%

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

   if 1.55e41 < x

   1. Initial program 88.0%

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

      1. *-commutative 88.0%

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

   3. Simplified 88.0%

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

      1. clear-num 88.0%

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

      2. inv-pow 88.0%

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

      3. fma-neg 88.0%

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

   6. Applied egg-rr 88.0%

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

      1. unpow-1 88.0%

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

      2. *-commutative 88.0%

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

      3. fma-neg 88.0%

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

      4. *-commutative 88.0%

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

   8. Simplified 88.0%

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

      1. div-inv 88.0%

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

      2. fma-neg 88.0%

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

   10. Applied egg-rr 88.0%

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

   11. Taylor expanded in z around inf 88.0%

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

   12. Taylor expanded in x around inf 100.0%

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

4. Final simplification 80.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -40000:\\ \;\;\;\;\frac{y \cdot \frac{z}{x} - x}{-1 - x}\\ \mathbf{elif}\;x \leq 1.55 \cdot 10^{+41}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.0042:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 3.2 \cdot 10^{+40}:\\ \;\;\;\;\frac{x + \frac{y}{t}}{x + 1}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.0042) 1.0 (if (<= x 3.2e+40) (/ (+ x (/ y t)) (+ x 1.0)) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.0042) {
		tmp = 1.0;
	} else if (x <= 3.2e+40) {
		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 (x <= (-0.0042d0)) then
        tmp = 1.0d0
    else if (x <= 3.2d+40) 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 (x <= -0.0042) {
		tmp = 1.0;
	} else if (x <= 3.2e+40) {
		tmp = (x + (y / t)) / (x + 1.0);
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -0.0042:
		tmp = 1.0
	elif x <= 3.2e+40:
		tmp = (x + (y / t)) / (x + 1.0)
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.0042)
		tmp = 1.0;
	elseif (x <= 3.2e+40)
		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 (x <= -0.0042)
		tmp = 1.0;
	elseif (x <= 3.2e+40)
		tmp = (x + (y / t)) / (x + 1.0);
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.0042], 1.0, If[LessEqual[x, 3.2e+40], 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 x < -0.00419999999999999974 or 3.19999999999999981e40 < 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 87.6%

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

      2. inv-pow 87.6%

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

      3. fma-neg 87.6%

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

   6. Applied egg-rr 87.6%

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

      1. unpow-1 87.6%

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

      2. *-commutative 87.6%

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

      3. fma-neg 87.6%

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

      4. *-commutative 87.6%

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

   8. Simplified 87.6%

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

      1. div-inv 87.6%

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

      2. fma-neg 87.6%

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

   10. Applied egg-rr 87.6%

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

   11. Taylor expanded in z around inf 86.2%

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

   12. Taylor expanded in x around inf 94.6%

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

   if -0.00419999999999999974 < x < 3.19999999999999981e40

   1. Initial program 91.6%

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

      1. *-commutative 91.6%

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

   3. Simplified 91.6%

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

   5. Taylor expanded in z around inf 67.1%

      \[\leadsto \frac{x + \color{blue}{\frac{y}{t}}}{x + 1} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 67.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.05 \cdot 10^{-9}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 1.5 \cdot 10^{-49}:\\ \;\;\;\;\frac{y}{t \cdot \left(x + 1\right)}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.05e-9) 1.0 (if (<= x 1.5e-49) (/ y (* t (+ x 1.0))) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.05e-9) {
		tmp = 1.0;
	} else if (x <= 1.5e-49) {
		tmp = 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 (x <= (-1.05d-9)) then
        tmp = 1.0d0
    else if (x <= 1.5d-49) then
        tmp = 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 (x <= -1.05e-9) {
		tmp = 1.0;
	} else if (x <= 1.5e-49) {
		tmp = y / (t * (x + 1.0));
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.05e-9:
		tmp = 1.0
	elif x <= 1.5e-49:
		tmp = y / (t * (x + 1.0))
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.05e-9)
		tmp = 1.0;
	elseif (x <= 1.5e-49)
		tmp = Float64(y / Float64(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 (x <= -1.05e-9)
		tmp = 1.0;
	elseif (x <= 1.5e-49)
		tmp = y / (t * (x + 1.0));
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.05e-9], 1.0, If[LessEqual[x, 1.5e-49], N[(y / N[(t * N[(x + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -1.0500000000000001e-9 or 1.5e-49 < x

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

   3. Simplified 88.4%

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

      1. clear-num 88.4%

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

      2. inv-pow 88.4%

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

      3. fma-neg 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. unpow-1 88.4%

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

      2. *-commutative 88.4%

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

      3. fma-neg 88.4%

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

      4. *-commutative 88.4%

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

   8. Simplified 88.4%

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

      1. div-inv 88.4%

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

      2. fma-neg 88.4%

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

   10. Applied egg-rr 88.4%

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

   11. Taylor expanded in z around inf 84.1%

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

   12. Taylor expanded in x around inf 88.2%

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

   if -1.0500000000000001e-9 < x < 1.5e-49

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in t around inf 78.5%

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

   6. Taylor expanded in y around inf 49.4%

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

      1. +-commutative 49.4%

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

   8. Simplified 49.4%

      \[\leadsto \color{blue}{\frac{y}{t \cdot \left(x + 1\right)}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 9: 67.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -9 \cdot 10^{-10}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-50}:\\ \;\;\;\;\frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -9e-10) 1.0 (if (<= x 7.5e-50) (/ y t) 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -9e-10) {
		tmp = 1.0;
	} else if (x <= 7.5e-50) {
		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 <= (-9d-10)) then
        tmp = 1.0d0
    else if (x <= 7.5d-50) 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 <= -9e-10) {
		tmp = 1.0;
	} else if (x <= 7.5e-50) {
		tmp = y / t;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -9e-10:
		tmp = 1.0
	elif x <= 7.5e-50:
		tmp = y / t
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -9e-10)
		tmp = 1.0;
	elseif (x <= 7.5e-50)
		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 <= -9e-10)
		tmp = 1.0;
	elseif (x <= 7.5e-50)
		tmp = y / t;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -9e-10], 1.0, If[LessEqual[x, 7.5e-50], N[(y / t), $MachinePrecision], 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -8.9999999999999999e-10 or 7.5e-50 < x

   1. Initial program 88.4%

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

      1. *-commutative 88.4%

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

   3. Simplified 88.4%

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

      1. clear-num 88.4%

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

      2. inv-pow 88.4%

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

      3. fma-neg 88.4%

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

   6. Applied egg-rr 88.4%

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

      1. unpow-1 88.4%

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

      2. *-commutative 88.4%

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

      3. fma-neg 88.4%

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

      4. *-commutative 88.4%

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

   8. Simplified 88.4%

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

      1. div-inv 88.4%

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

      2. fma-neg 88.4%

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

   10. Applied egg-rr 88.4%

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

   11. Taylor expanded in z around inf 84.1%

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

   12. Taylor expanded in x around inf 88.2%

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

   if -8.9999999999999999e-10 < x < 7.5e-50

   1. Initial program 91.1%

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

      1. *-commutative 91.1%

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

   3. Simplified 91.1%

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

   5. Taylor expanded in t around inf 78.5%

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

   6. Taylor expanded in y around inf 49.4%

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

      1. +-commutative 49.4%

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

   8. Simplified 49.4%

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

   9. Taylor expanded in x around 0 49.4%

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

Alternative 10: 55.7% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -4.4 \cdot 10^{-138}:\\ \;\;\;\;1\\ \mathbf{elif}\;x \leq 4.7 \cdot 10^{-117}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -4.4e-138) 1.0 (if (<= x 4.7e-117) x 1.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -4.4e-138) {
		tmp = 1.0;
	} else if (x <= 4.7e-117) {
		tmp = x;
	} 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 <= (-4.4d-138)) then
        tmp = 1.0d0
    else if (x <= 4.7d-117) then
        tmp = x
    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 <= -4.4e-138) {
		tmp = 1.0;
	} else if (x <= 4.7e-117) {
		tmp = x;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -4.4e-138:
		tmp = 1.0
	elif x <= 4.7e-117:
		tmp = x
	else:
		tmp = 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -4.4e-138)
		tmp = 1.0;
	elseif (x <= 4.7e-117)
		tmp = x;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -4.4e-138)
		tmp = 1.0;
	elseif (x <= 4.7e-117)
		tmp = x;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -4.4e-138], 1.0, If[LessEqual[x, 4.7e-117], x, 1.0]]

Derivation

1. Split input into 2 regimes

2. if x < -4.3999999999999998e-138 or 4.70000000000000009e-117 < 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. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 88.8%

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

      2. inv-pow 88.8%

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

      3. fma-neg 88.8%

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

   6. Applied egg-rr 88.8%

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

      1. unpow-1 88.8%

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

      2. *-commutative 88.8%

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

      3. fma-neg 88.8%

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

      4. *-commutative 88.8%

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

   8. Simplified 88.8%

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

      1. div-inv 88.7%

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

      2. fma-neg 88.7%

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

   10. Applied egg-rr 88.7%

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

   11. Taylor expanded in z around inf 79.6%

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

   12. Taylor expanded in x around inf 76.0%

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

   if -4.3999999999999998e-138 < x < 4.70000000000000009e-117

   1. Initial program 91.8%

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

      1. *-commutative 91.8%

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

   3. Simplified 91.8%

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

   5. Taylor expanded in t around inf 22.7%

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

      1. +-commutative 22.7%

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

   7. Simplified 22.7%

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

   8. Taylor expanded in x around 0 22.7%

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

      1. neg-mul-1 22.7%

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

   10. Simplified 22.7%

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

   11. Taylor expanded in x around 0 22.7%

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

Alternative 11: 53.5% 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 89.6%

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

   1. *-commutative 89.6%

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

3. Simplified 89.6%

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

   1. clear-num 89.6%

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

   2. inv-pow 89.6%

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

   3. fma-neg 89.6%

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

6. Applied egg-rr 89.6%

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

   1. unpow-1 89.6%

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

   2. *-commutative 89.6%

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

   3. fma-neg 89.6%

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

   4. *-commutative 89.6%

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

8. Simplified 89.6%

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

   1. div-inv 89.5%

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

   2. fma-neg 89.5%

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

10. Applied egg-rr 89.5%

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

11. Taylor expanded in z around inf 77.0%

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

12. Taylor expanded in x around inf 56.6%

    \[\leadsto \color{blue}{1} \]
13. Add Preprocessing

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 2024101 
(FPCore (x y z t)
  :name "Diagrams.Trail:splitAtParam  from diagrams-lib-1.3.0.3, A"
  :precision binary64

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

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