Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B

Percentage Accurate: 96.0% → 97.9%

Time: 9.2s

Alternatives: 8

Speedup: 0.6×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 5e+297) (/ x (fma (- z) t y)) (/ (/ x (- z)) t)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 5e+297) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = (x / -z) / t;
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 5e+297)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(x / Float64(-z)) / t);
	end
	return tmp
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 5e+297], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 4.9999999999999998e297

   1. Initial program 98.6%

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

      1. sub-neg 98.6%

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

      2. +-commutative 98.6%

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

      3. distribute-lft-neg-in 98.6%

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

      4. fma-def 98.6%

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

   3. Applied egg-rr 98.6%

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

   if 4.9999999999999998e297 < (*.f64 z t)

   1. Initial program 72.6%

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

   2. Taylor expanded in y around 0 72.6%

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

      1. associate-*r/ 72.6%

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

      2. neg-mul-1 72.6%

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

   4. Simplified 72.6%

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

      1. neg-mul-1 72.6%

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

      2. *-commutative 72.6%

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

      3. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.7%

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

      2. frac-2neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-*l/ 99.7%

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

      5. *-un-lft-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \]

Alternative 2: 79.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{+78}:\\ \;\;\;\;x \cdot \frac{-1}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e-5)
   (/ (- (/ x t)) z)
   (if (<= (* z t) 2e-108)
     (/ x y)
     (if (<= (* z t) 1e+78) (* x (/ -1.0 (* z t))) (/ (/ x (- z)) t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e-5) {
		tmp = -(x / t) / z;
	} else if ((z * t) <= 2e-108) {
		tmp = x / y;
	} else if ((z * t) <= 1e+78) {
		tmp = x * (-1.0 / (z * t));
	} else {
		tmp = (x / -z) / t;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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 * t) <= (-2d-5)) then
        tmp = -(x / t) / z
    else if ((z * t) <= 2d-108) then
        tmp = x / y
    else if ((z * t) <= 1d+78) then
        tmp = x * ((-1.0d0) / (z * t))
    else
        tmp = (x / -z) / t
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e-5) {
		tmp = -(x / t) / z;
	} else if ((z * t) <= 2e-108) {
		tmp = x / y;
	} else if ((z * t) <= 1e+78) {
		tmp = x * (-1.0 / (z * t));
	} else {
		tmp = (x / -z) / t;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e-5:
		tmp = -(x / t) / z
	elif (z * t) <= 2e-108:
		tmp = x / y
	elif (z * t) <= 1e+78:
		tmp = x * (-1.0 / (z * t))
	else:
		tmp = (x / -z) / t
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e-5)
		tmp = Float64(Float64(-Float64(x / t)) / z);
	elseif (Float64(z * t) <= 2e-108)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 1e+78)
		tmp = Float64(x * Float64(-1.0 / Float64(z * t)));
	else
		tmp = Float64(Float64(x / Float64(-z)) / t);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -2e-5)
		tmp = -(x / t) / z;
	elseif ((z * t) <= 2e-108)
		tmp = x / y;
	elseif ((z * t) <= 1e+78)
		tmp = x * (-1.0 / (z * t));
	else
		tmp = (x / -z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e-5], N[((-N[(x / t), $MachinePrecision]) / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-108], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+78], N[(x * N[(-1.0 / N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -2.00000000000000016e-5

   1. Initial program 94.8%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

   3. Applied egg-rr 94.7%

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

   4. Taylor expanded in y around 0 77.1%

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

      1. mul-1-neg 77.1%

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

      2. associate-/r* 75.7%

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

      3. distribute-neg-frac 75.7%

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

   6. Simplified 75.7%

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

   if -2.00000000000000016e-5 < (*.f64 z t) < 2.00000000000000008e-108

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 89.6%

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

   if 2.00000000000000008e-108 < (*.f64 z t) < 1.00000000000000001e78

   1. Initial program 99.6%

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

      1. clear-num 99.3%

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

      2. associate-/r/ 99.5%

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

   3. Applied egg-rr 99.5%

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

   4. Taylor expanded in y around 0 58.4%

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

   if 1.00000000000000001e78 < (*.f64 z t)

   1. Initial program 90.8%

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

   2. Taylor expanded in y around 0 79.7%

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

      1. associate-*r/ 79.7%

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

      2. neg-mul-1 79.7%

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

   4. Simplified 79.7%

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

      1. neg-mul-1 79.7%

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

      2. *-commutative 79.7%

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

      3. times-frac 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. associate-*r/ 87.0%

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

      2. frac-2neg 87.0%

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

      3. metadata-eval 87.0%

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

      4. associate-*l/ 87.0%

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

      5. *-un-lft-identity 87.0%

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

   8. Applied egg-rr 87.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{-z}}{t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{+78}:\\ \;\;\;\;x \cdot \frac{-1}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \]

Alternative 3: 79.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{-\frac{x}{t}}{z}\\ \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-5}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- (/ x t)) z)))
   (if (<= (* z t) -2e-5)
     t_1
     (if (<= (* z t) 2e-108)
       (/ x y)
       (if (<= (* z t) 5e+285) (/ (- x) (* z t)) t_1)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double t_1 = -(x / t) / z;
	double tmp;
	if ((z * t) <= -2e-5) {
		tmp = t_1;
	} else if ((z * t) <= 2e-108) {
		tmp = x / y;
	} else if ((z * t) <= 5e+285) {
		tmp = -x / (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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 / t) / z
    if ((z * t) <= (-2d-5)) then
        tmp = t_1
    else if ((z * t) <= 2d-108) then
        tmp = x / y
    else if ((z * t) <= 5d+285) then
        tmp = -x / (z * t)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double t_1 = -(x / t) / z;
	double tmp;
	if ((z * t) <= -2e-5) {
		tmp = t_1;
	} else if ((z * t) <= 2e-108) {
		tmp = x / y;
	} else if ((z * t) <= 5e+285) {
		tmp = -x / (z * t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	t_1 = -(x / t) / z
	tmp = 0
	if (z * t) <= -2e-5:
		tmp = t_1
	elif (z * t) <= 2e-108:
		tmp = x / y
	elif (z * t) <= 5e+285:
		tmp = -x / (z * t)
	else:
		tmp = t_1
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(-Float64(x / t)) / z)
	tmp = 0.0
	if (Float64(z * t) <= -2e-5)
		tmp = t_1;
	elseif (Float64(z * t) <= 2e-108)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 5e+285)
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = -(x / t) / z;
	tmp = 0.0;
	if ((z * t) <= -2e-5)
		tmp = t_1;
	elseif ((z * t) <= 2e-108)
		tmp = x / y;
	elseif ((z * t) <= 5e+285)
		tmp = -x / (z * t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-N[(x / t), $MachinePrecision]) / z), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -2e-5], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 2e-108], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+285], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.00000000000000016e-5 or 5.00000000000000016e285 < (*.f64 z t)

   1. Initial program 90.1%

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

      1. clear-num 90.0%

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

      2. inv-pow 90.0%

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

   3. Applied egg-rr 90.0%

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

   4. Taylor expanded in y around 0 75.6%

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

      1. mul-1-neg 75.6%

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

      2. associate-/r* 80.5%

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

      3. distribute-neg-frac 80.5%

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

   6. Simplified 80.5%

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

   if -2.00000000000000016e-5 < (*.f64 z t) < 2.00000000000000008e-108

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 89.6%

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

   if 2.00000000000000008e-108 < (*.f64 z t) < 5.00000000000000016e285

   1. Initial program 99.7%

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

   2. Taylor expanded in y around 0 69.3%

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

      1. associate-*r/ 69.3%

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

      2. neg-mul-1 69.3%

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

   4. Simplified 69.3%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+285}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \end{array} \]

Alternative 4: 79.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{+78}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e-5)
   (/ (- (/ x t)) z)
   (if (<= (* z t) 2e-108)
     (/ x y)
     (if (<= (* z t) 1e+78) (/ (- x) (* z t)) (/ (/ x (- z)) t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e-5) {
		tmp = -(x / t) / z;
	} else if ((z * t) <= 2e-108) {
		tmp = x / y;
	} else if ((z * t) <= 1e+78) {
		tmp = -x / (z * t);
	} else {
		tmp = (x / -z) / t;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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 * t) <= (-2d-5)) then
        tmp = -(x / t) / z
    else if ((z * t) <= 2d-108) then
        tmp = x / y
    else if ((z * t) <= 1d+78) then
        tmp = -x / (z * t)
    else
        tmp = (x / -z) / t
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e-5) {
		tmp = -(x / t) / z;
	} else if ((z * t) <= 2e-108) {
		tmp = x / y;
	} else if ((z * t) <= 1e+78) {
		tmp = -x / (z * t);
	} else {
		tmp = (x / -z) / t;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e-5:
		tmp = -(x / t) / z
	elif (z * t) <= 2e-108:
		tmp = x / y
	elif (z * t) <= 1e+78:
		tmp = -x / (z * t)
	else:
		tmp = (x / -z) / t
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e-5)
		tmp = Float64(Float64(-Float64(x / t)) / z);
	elseif (Float64(z * t) <= 2e-108)
		tmp = Float64(x / y);
	elseif (Float64(z * t) <= 1e+78)
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(Float64(x / Float64(-z)) / t);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -2e-5)
		tmp = -(x / t) / z;
	elseif ((z * t) <= 2e-108)
		tmp = x / y;
	elseif ((z * t) <= 1e+78)
		tmp = -x / (z * t);
	else
		tmp = (x / -z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], -2e-5], N[((-N[(x / t), $MachinePrecision]) / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e-108], N[(x / y), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 1e+78], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if (*.f64 z t) < -2.00000000000000016e-5

   1. Initial program 94.8%

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

      1. clear-num 94.7%

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

      2. inv-pow 94.7%

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

   3. Applied egg-rr 94.7%

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

   4. Taylor expanded in y around 0 77.1%

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

      1. mul-1-neg 77.1%

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

      2. associate-/r* 75.7%

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

      3. distribute-neg-frac 75.7%

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

   6. Simplified 75.7%

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

   if -2.00000000000000016e-5 < (*.f64 z t) < 2.00000000000000008e-108

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 89.6%

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

   if 2.00000000000000008e-108 < (*.f64 z t) < 1.00000000000000001e78

   1. Initial program 99.6%

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

   2. Taylor expanded in y around 0 58.3%

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

      1. associate-*r/ 58.3%

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

      2. neg-mul-1 58.3%

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

   4. Simplified 58.3%

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

   if 1.00000000000000001e78 < (*.f64 z t)

   1. Initial program 90.8%

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

   2. Taylor expanded in y around 0 79.7%

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

      1. associate-*r/ 79.7%

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

      2. neg-mul-1 79.7%

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

   4. Simplified 79.7%

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

      1. neg-mul-1 79.7%

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

      2. *-commutative 79.7%

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

      3. times-frac 83.7%

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

   6. Applied egg-rr 83.7%

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

      1. associate-*r/ 87.0%

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

      2. frac-2neg 87.0%

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

      3. metadata-eval 87.0%

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

      4. associate-*l/ 87.0%

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

      5. *-un-lft-identity 87.0%

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

   8. Applied egg-rr 87.0%

      \[\leadsto \color{blue}{\frac{\frac{x}{-z}}{t}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-5}:\\ \;\;\;\;\frac{-\frac{x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{-108}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \cdot t \leq 10^{+78}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \]

Alternative 5: 76.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-5} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -2e-5) (not (<= (* z t) 2e-108)))
   (/ (- x) (* z t))
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e-5) || !((z * t) <= 2e-108)) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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 * t) <= (-2d-5)) .or. (.not. ((z * t) <= 2d-108))) then
        tmp = -x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e-5) || !((z * t) <= 2e-108)) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -2e-5) or not ((z * t) <= 2e-108):
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -2e-5) || !(Float64(z * t) <= 2e-108))
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -2e-5) || ~(((z * t) <= 2e-108)))
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e-5], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e-108]], $MachinePrecision]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.00000000000000016e-5 or 2.00000000000000008e-108 < (*.f64 z t)

   1. Initial program 94.9%

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

   2. Taylor expanded in y around 0 72.4%

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

      1. associate-*r/ 72.4%

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

      2. neg-mul-1 72.4%

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

   4. Simplified 72.4%

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

   if -2.00000000000000016e-5 < (*.f64 z t) < 2.00000000000000008e-108

   1. Initial program 100.0%

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

   2. Taylor expanded in y around inf 89.6%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{-5} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{-108}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 63.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+186} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -5e+186) (not (<= (* z t) 2e+88))) (/ x (* z t)) (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+186) || !((z * t) <= 2e+88)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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 * t) <= (-5d+186)) .or. (.not. ((z * t) <= 2d+88))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+186) || !((z * t) <= 2e+88)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+186) or not ((z * t) <= 2e+88):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+186) || !(Float64(z * t) <= 2e+88))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -5e+186) || ~(((z * t) <= 2e+88)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+186], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+88]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.99999999999999954e186 or 1.99999999999999992e88 < (*.f64 z t)

   1. Initial program 89.2%

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

   2. Taylor expanded in y around 0 82.6%

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

      1. associate-*r/ 82.6%

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

      2. neg-mul-1 82.6%

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

   4. Simplified 82.6%

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

      1. expm1-log1p-u 76.8%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-x}{t \cdot z}\right)\right)} \]

      2. expm1-udef 59.3%

         \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{-x}{t \cdot z}\right)} - 1} \]

      3. add-sqr-sqrt 25.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{-x} \cdot \sqrt{-x}}}{t \cdot z}\right)} - 1 \]

      4. sqrt-unprod 56.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{\left(-x\right) \cdot \left(-x\right)}}}{t \cdot z}\right)} - 1 \]

      5. sqr-neg 56.9%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\sqrt{\color{blue}{x \cdot x}}}{t \cdot z}\right)} - 1 \]

      6. sqrt-unprod 31.8%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{\sqrt{x} \cdot \sqrt{x}}}{t \cdot z}\right)} - 1 \]

      7. add-sqr-sqrt 56.7%

         \[\leadsto e^{\mathsf{log1p}\left(\frac{\color{blue}{x}}{t \cdot z}\right)} - 1 \]

      8. associate-/r* 56.7%

         \[\leadsto e^{\mathsf{log1p}\left(\color{blue}{\frac{\frac{x}{t}}{z}}\right)} - 1 \]

   6. Applied egg-rr 56.7%

      \[\leadsto \color{blue}{e^{\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)} - 1} \]
   7. Step-by-step derivation

      1. expm1-def 55.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{\frac{x}{t}}{z}\right)\right)} \]

      2. expm1-log1p 55.0%

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

      3. associate-/r* 55.2%

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

   8. Simplified 55.2%

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

   if -4.99999999999999954e186 < (*.f64 z t) < 1.99999999999999992e88

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 67.4%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+186} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+88}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 97.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) 5e+297) (/ x (- y (* z t))) (/ (/ x (- z)) t)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 5e+297) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / -z) / t;
	}
	return tmp;
}

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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 * t) <= 5d+297) then
        tmp = x / (y - (z * t))
    else
        tmp = (x / -z) / t
    end if
    code = tmp
end function

Java

assert z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 5e+297) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / -z) / t;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= 5e+297:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / -z) / t
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= 5e+297)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / Float64(-z)) / t);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= 5e+297)
		tmp = x / (y - (z * t));
	else
		tmp = (x / -z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], 5e+297], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 4.9999999999999998e297

   1. Initial program 98.6%

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

   if 4.9999999999999998e297 < (*.f64 z t)

   1. Initial program 72.6%

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

   2. Taylor expanded in y around 0 72.6%

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

      1. associate-*r/ 72.6%

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

      2. neg-mul-1 72.6%

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

   4. Simplified 72.6%

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

      1. neg-mul-1 72.6%

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

      2. *-commutative 72.6%

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

      3. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*r/ 99.7%

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

      2. frac-2neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. associate-*l/ 99.7%

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

      5. *-un-lft-identity 99.7%

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

   8. Applied egg-rr 99.7%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+297}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \]

Alternative 8: 54.9% accurate, 2.3× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \frac{x}{y} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t) :precision binary64 (/ x y))

C

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

Fortran

NOTE: z and t should be sorted in increasing order before calling this function.
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
end function

Java

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

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	return x / y

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	return Float64(x / y)
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / y;
end

Wolfram

NOTE: z and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := N[(x / y), $MachinePrecision]

Derivation

1. Initial program 96.9%

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

2. Taylor expanded in y around inf 54.6%

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

3. Final simplification 54.6%

   \[\leadsto \frac{x}{y} \]

Developer target: 96.4% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{1}{\frac{y}{x} - \frac{z}{x} \cdot t}\\ \mathbf{if}\;x < -1.618195973607049 \cdot 10^{+50}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;x < 2.1378306434876444 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 1.0 (- (/ y x) (* (/ z x) t)))))
   (if (< x -1.618195973607049e+50)
     t_1
     (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = 1.0d0 / ((y / x) - ((z / x) * t))
    if (x < (-1.618195973607049d+50)) then
        tmp = t_1
    else if (x < 2.1378306434876444d+131) then
        tmp = x / (y - (z * t))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 1.0 / ((y / x) - ((z / x) * t));
	double tmp;
	if (x < -1.618195973607049e+50) {
		tmp = t_1;
	} else if (x < 2.1378306434876444e+131) {
		tmp = x / (y - (z * t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 1.0 / ((y / x) - ((z / x) * t))
	tmp = 0
	if x < -1.618195973607049e+50:
		tmp = t_1
	elif x < 2.1378306434876444e+131:
		tmp = x / (y - (z * t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(1.0 / Float64(Float64(y / x) - Float64(Float64(z / x) * t)))
	tmp = 0.0
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 1.0 / ((y / x) - ((z / x) * t));
	tmp = 0.0;
	if (x < -1.618195973607049e+50)
		tmp = t_1;
	elseif (x < 2.1378306434876444e+131)
		tmp = x / (y - (z * t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(1.0 / N[(N[(y / x), $MachinePrecision] - N[(N[(z / x), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[x, -1.618195973607049e+50], t$95$1, If[Less[x, 2.1378306434876444e+131], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2023229 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :herbie-target
  (if (< x -1.618195973607049e+50) (/ 1.0 (- (/ y x) (* (/ z x) t))) (if (< x 2.1378306434876444e+131) (/ x (- y (* z t))) (/ 1.0 (- (/ y x) (* (/ z x) t)))))

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