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

Percentage Accurate: 95.7% → 98.9%

Time: 13.8s

Alternatives: 8

Speedup: 0.5×

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: 95.7% 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: 98.9% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \end{array} \end{array} \]

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+303)
   (/ (/ x t) (- z))
   (if (<= (* z t) 2e+151) (/ x (fma z (- t) y)) (* (/ x t) (/ -1.0 z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -5e+303) {
		tmp = (x / t) / -z;
	} else if ((z * t) <= 2e+151) {
		tmp = x / fma(z, -t, y);
	} else {
		tmp = (x / t) * (-1.0 / z);
	}
	return tmp;
}

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -5e+303)
		tmp = Float64(Float64(x / t) / Float64(-z));
	elseif (Float64(z * t) <= 2e+151)
		tmp = Float64(x / fma(z, Float64(-t), y));
	else
		tmp = Float64(Float64(x / t) * Float64(-1.0 / z));
	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+303], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+151], N[(x / N[(z * (-t) + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.9999999999999997e303

   1. Initial program 63.5%

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

   2. Taylor expanded in y around 0 63.5%

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

      1. associate-*r/ 63.5%

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

      2. neg-mul-1 63.5%

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

      3. *-commutative 63.5%

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

      4. associate-/r* 100.0%

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

   4. Simplified 100.0%

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

      1. associate-/l/ 63.5%

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

      2. neg-mul-1 63.5%

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

      3. *-commutative 63.5%

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

      4. times-frac 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. frac-2neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

   if -4.9999999999999997e303 < (*.f64 z t) < 2.00000000000000003e151

   1. Initial program 99.9%

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

      1. cancel-sign-sub-inv 99.9%

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

      2. +-commutative 99.9%

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

      3. distribute-lft-neg-out 99.9%

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

      4. distribute-rgt-neg-out 99.9%

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

      5. fma-def 99.9%

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

   3. Simplified 99.9%

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

   if 2.00000000000000003e151 < (*.f64 z t)

   1. Initial program 66.1%

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

   2. Taylor expanded in y around 0 66.1%

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

      1. associate-*r/ 66.1%

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

      2. neg-mul-1 66.1%

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

      3. *-commutative 66.1%

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

      4. associate-/r* 99.8%

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

   4. Simplified 99.8%

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

      1. associate-/l/ 66.1%

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

      2. neg-mul-1 66.1%

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

      3. *-commutative 66.1%

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

      4. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \end{array} \]

Alternative 2: 98.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \end{array} \end{array} \]

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+303)
   (/ (/ x t) (- z))
   (if (<= (* z t) 2e+151) (/ x (- y (* z t))) (* (/ x t) (/ -1.0 z)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -5e+303) {
		tmp = (x / t) / -z;
	} else if ((z * t) <= 2e+151) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) * (-1.0 / z);
	}
	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+303)) then
        tmp = (x / t) / -z
    else if ((z * t) <= 2d+151) then
        tmp = x / (y - (z * t))
    else
        tmp = (x / t) * ((-1.0d0) / z)
    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+303) {
		tmp = (x / t) / -z;
	} else if ((z * t) <= 2e+151) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / t) * (-1.0 / z);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -5e+303:
		tmp = (x / t) / -z
	elif (z * t) <= 2e+151:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / t) * (-1.0 / z)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -5e+303)
		tmp = Float64(Float64(x / t) / Float64(-z));
	elseif (Float64(z * t) <= 2e+151)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / t) * Float64(-1.0 / z));
	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+303)
		tmp = (x / t) / -z;
	elseif ((z * t) <= 2e+151)
		tmp = x / (y - (z * t));
	else
		tmp = (x / t) * (-1.0 / z);
	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+303], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+151], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -4.9999999999999997e303

   1. Initial program 63.5%

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

   2. Taylor expanded in y around 0 63.5%

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

      1. associate-*r/ 63.5%

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

      2. neg-mul-1 63.5%

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

      3. *-commutative 63.5%

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

      4. associate-/r* 100.0%

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

   4. Simplified 100.0%

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

      1. associate-/l/ 63.5%

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

      2. neg-mul-1 63.5%

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

      3. *-commutative 63.5%

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

      4. times-frac 99.7%

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

   6. Applied egg-rr 99.7%

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

      1. *-commutative 99.7%

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

      2. frac-2neg 99.7%

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

      3. metadata-eval 99.7%

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

      4. un-div-inv 99.8%

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

   8. Applied egg-rr 99.8%

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

   if -4.9999999999999997e303 < (*.f64 z t) < 2.00000000000000003e151

   1. Initial program 99.9%

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

   if 2.00000000000000003e151 < (*.f64 z t)

   1. Initial program 66.1%

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

   2. Taylor expanded in y around 0 66.1%

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

      1. associate-*r/ 66.1%

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

      2. neg-mul-1 66.1%

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

      3. *-commutative 66.1%

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

      4. associate-/r* 99.8%

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

   4. Simplified 99.8%

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

      1. associate-/l/ 66.1%

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

      2. neg-mul-1 66.1%

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

      3. *-commutative 66.1%

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

      4. times-frac 99.8%

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

   6. Applied egg-rr 99.8%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+303}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+151}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \end{array} \]

Alternative 3: 70.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.5e-25)
   (/ x y)
   (if (<= y 3.5e-99) (/ -1.0 (* z (/ t x))) (/ x y))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.5e-25) {
		tmp = x / y;
	} else if (y <= 3.5e-99) {
		tmp = -1.0 / (z * (t / x));
	} 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 (y <= (-4.5d-25)) then
        tmp = x / y
    else if (y <= 3.5d-99) then
        tmp = (-1.0d0) / (z * (t / x))
    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 (y <= -4.5e-25) {
		tmp = x / y;
	} else if (y <= 3.5e-99) {
		tmp = -1.0 / (z * (t / x));
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -4.5e-25:
		tmp = x / y
	elif y <= 3.5e-99:
		tmp = -1.0 / (z * (t / x))
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.5e-25)
		tmp = Float64(x / y);
	elseif (y <= 3.5e-99)
		tmp = Float64(-1.0 / Float64(z * Float64(t / x)));
	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 (y <= -4.5e-25)
		tmp = x / y;
	elseif (y <= 3.5e-99)
		tmp = -1.0 / (z * (t / x));
	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[LessEqual[y, -4.5e-25], N[(x / y), $MachinePrecision], If[LessEqual[y, 3.5e-99], N[(-1.0 / N[(z * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.5000000000000001e-25 or 3.4999999999999999e-99 < y

   1. Initial program 95.0%

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

   2. Taylor expanded in y around inf 75.2%

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

   if -4.5000000000000001e-25 < y < 3.4999999999999999e-99

   1. Initial program 92.2%

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

   2. Taylor expanded in y around 0 78.5%

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

      1. associate-*r/ 78.5%

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

      2. neg-mul-1 78.5%

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

      3. *-commutative 78.5%

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

      4. associate-/r* 81.6%

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

   4. Simplified 81.6%

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

      1. associate-/l/ 78.5%

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

      2. neg-mul-1 78.5%

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

      3. *-commutative 78.5%

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

      4. times-frac 80.2%

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

   6. Applied egg-rr 80.2%

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

      1. *-commutative 80.2%

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

      2. clear-num 80.1%

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

      3. frac-times 80.7%

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

      4. metadata-eval 80.7%

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

   8. Applied egg-rr 80.7%

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

4. Final simplification 77.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 3.5 \cdot 10^{-99}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 4: 73.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.5e-27) (/ x y) (if (<= y 5e-8) (/ (- x) (* z t)) (/ x y))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.5e-27) {
		tmp = x / y;
	} else if (y <= 5e-8) {
		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 (y <= (-1.5d-27)) then
        tmp = x / y
    else if (y <= 5d-8) 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 (y <= -1.5e-27) {
		tmp = x / y;
	} else if (y <= 5e-8) {
		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 y <= -1.5e-27:
		tmp = x / y
	elif y <= 5e-8:
		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 (y <= -1.5e-27)
		tmp = Float64(x / y);
	elseif (y <= 5e-8)
		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 (y <= -1.5e-27)
		tmp = x / y;
	elseif (y <= 5e-8)
		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[LessEqual[y, -1.5e-27], N[(x / y), $MachinePrecision], If[LessEqual[y, 5e-8], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.5000000000000001e-27 or 4.9999999999999998e-8 < y

   1. Initial program 94.4%

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

   2. Taylor expanded in y around inf 79.1%

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

   if -1.5000000000000001e-27 < y < 4.9999999999999998e-8

   1. Initial program 93.3%

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

   2. Taylor expanded in y around 0 75.4%

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

      1. associate-*r/ 75.4%

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

      2. neg-mul-1 75.4%

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

   4. Simplified 75.4%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 5 \cdot 10^{-8}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 72.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.8e-25) (/ x y) (if (<= y 8e-11) (/ (/ x t) (- z)) (/ x y))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.8e-25) {
		tmp = x / y;
	} else if (y <= 8e-11) {
		tmp = (x / t) / -z;
	} 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 (y <= (-6.8d-25)) then
        tmp = x / y
    else if (y <= 8d-11) then
        tmp = (x / t) / -z
    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 (y <= -6.8e-25) {
		tmp = x / y;
	} else if (y <= 8e-11) {
		tmp = (x / t) / -z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if y <= -6.8e-25:
		tmp = x / y
	elif y <= 8e-11:
		tmp = (x / t) / -z
	else:
		tmp = x / y
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.8e-25)
		tmp = Float64(x / y);
	elseif (y <= 8e-11)
		tmp = Float64(Float64(x / t) / Float64(-z));
	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 (y <= -6.8e-25)
		tmp = x / y;
	elseif (y <= 8e-11)
		tmp = (x / t) / -z;
	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[LessEqual[y, -6.8e-25], N[(x / y), $MachinePrecision], If[LessEqual[y, 8e-11], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], N[(x / y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.80000000000000003e-25 or 7.99999999999999952e-11 < y

   1. Initial program 94.4%

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

   2. Taylor expanded in y around inf 79.1%

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

   if -6.80000000000000003e-25 < y < 7.99999999999999952e-11

   1. Initial program 93.3%

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

   2. Taylor expanded in y around 0 75.4%

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

      1. associate-*r/ 75.4%

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

      2. neg-mul-1 75.4%

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

      3. *-commutative 75.4%

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

      4. associate-/r* 76.6%

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

   4. Simplified 76.6%

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

      1. associate-/l/ 75.4%

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

      2. neg-mul-1 75.4%

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

      3. *-commutative 75.4%

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

      4. times-frac 76.6%

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

   6. Applied egg-rr 76.6%

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

      1. *-commutative 76.6%

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

      2. frac-2neg 76.6%

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

      3. metadata-eval 76.6%

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

      4. un-div-inv 76.8%

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

   8. Applied egg-rr 76.8%

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

4. Final simplification 78.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.8 \cdot 10^{-25}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;y \leq 8 \cdot 10^{-11}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 57.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+215} \lor \neg \left(z \leq 2.95 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 -2e+215) (not (<= z 2.95e-33))) (/ x (* z t)) (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2e+215) || !(z <= 2.95e-33)) {
		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 <= (-2d+215)) .or. (.not. (z <= 2.95d-33))) 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 <= -2e+215) || !(z <= 2.95e-33)) {
		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 <= -2e+215) or not (z <= 2.95e-33):
		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 ((z <= -2e+215) || !(z <= 2.95e-33))
		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 <= -2e+215) || ~((z <= 2.95e-33)))
		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[z, -2e+215], N[Not[LessEqual[z, 2.95e-33]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999981e215 or 2.94999999999999993e-33 < z

   1. Initial program 85.5%

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

   2. Taylor expanded in y around 0 61.1%

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

      1. associate-*r/ 61.1%

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

      2. neg-mul-1 61.1%

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

      3. *-commutative 61.1%

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

      4. associate-/r* 70.6%

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

   4. Simplified 70.6%

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

      1. expm1-log1p-u 64.4%

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

      2. expm1-udef 33.4%

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

      3. associate-/l/ 33.4%

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

      4. add-sqr-sqrt 16.1%

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

      5. *-commutative 16.1%

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

      6. sqrt-unprod 23.5%

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

      7. sqr-neg 23.5%

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

      8. sqrt-unprod 12.0%

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

      9. add-sqr-sqrt 24.3%

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

   6. Applied egg-rr 24.3%

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

      1. expm1-def 22.7%

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

      2. expm1-log1p 23.0%

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

      3. *-commutative 23.0%

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

   8. Simplified 23.0%

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

   if -1.99999999999999981e215 < z < 2.94999999999999993e-33

   1. Initial program 98.7%

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

   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 51.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{+215} \lor \neg \left(z \leq 2.95 \cdot 10^{-33}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 55.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+207}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t 7.2e+207) (/ x y) (/ (/ x t) z)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 7.2e+207) {
		tmp = x / y;
	} else {
		tmp = (x / t) / z;
	}
	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 (t <= 7.2d+207) then
        tmp = x / y
    else
        tmp = (x / t) / z
    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 (t <= 7.2e+207) {
		tmp = x / y;
	} else {
		tmp = (x / t) / z;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= 7.2e+207:
		tmp = x / y
	else:
		tmp = (x / t) / z
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= 7.2e+207)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / z);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 7.2e+207)
		tmp = x / y;
	else
		tmp = (x / t) / z;
	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[t, 7.2e+207], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7.20000000000000028e207

   1. Initial program 94.9%

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

   2. Taylor expanded in y around inf 55.9%

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

   if 7.20000000000000028e207 < t

   1. Initial program 84.4%

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

   2. Taylor expanded in y around 0 64.9%

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

      1. associate-*r/ 64.9%

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

      2. neg-mul-1 64.9%

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

      3. *-commutative 64.9%

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

      4. associate-/r* 76.5%

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

   4. Simplified 76.5%

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

      1. expm1-log1p-u 76.1%

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

      2. expm1-udef 64.2%

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

      3. associate-/l/ 68.0%

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

      4. add-sqr-sqrt 47.0%

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

      5. *-commutative 47.0%

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

      6. sqrt-unprod 63.5%

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

      7. sqr-neg 63.5%

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

      8. sqrt-unprod 21.0%

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

      9. add-sqr-sqrt 60.1%

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

   6. Applied egg-rr 60.1%

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

      1. expm1-def 48.1%

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

      2. expm1-log1p 48.2%

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

      3. *-commutative 48.2%

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

   8. Simplified 48.2%

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

      1. add-sqr-sqrt 17.0%

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

      2. sqrt-unprod 60.0%

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

      3. sqr-neg 60.0%

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

      4. sqrt-unprod 47.7%

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

      5. add-sqr-sqrt 64.9%

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

      6. expm1-log1p-u 64.4%

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

      7. expm1-udef 68.0%

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

      8. add-sqr-sqrt 47.0%

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

      9. sqrt-unprod 63.5%

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

      10. sqr-neg 63.5%

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

      11. sqrt-unprod 21.0%

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

      12. add-sqr-sqrt 60.1%

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

      13. *-commutative 60.1%

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

   10. Applied egg-rr 60.1%

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

       1. expm1-def 48.1%

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

       2. expm1-log1p 48.2%

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

       3. associate-/l/ 59.8%

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

   12. Simplified 59.8%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 7.2 \cdot 10^{+207}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 8: 52.7% 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 93.9%

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

2. Taylor expanded in y around inf 54.5%

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

3. Final simplification 54.5%

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

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 2023275 
(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))))