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

Percentage Accurate: 96.0% → 98.1%

Time: 9.8s

Alternatives: 7

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: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \end{array} \]

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) 2e+269) (/ 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) <= 2e+269) {
		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) <= 2d+269) 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) <= 2e+269) {
		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) <= 2e+269:
		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) <= 2e+269)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / z) / Float64(-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+269)
		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], 2e+269], 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) < 2.0000000000000001e269

   1. Initial program 98.6%

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

   if 2.0000000000000001e269 < (*.f64 z t)

   1. Initial program 61.9%

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

      1. *-un-lft-identity 61.9%

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

      2. *-commutative 61.9%

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

      3. *-commutative 61.9%

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

      4. add-sqr-sqrt 37.2%

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

      5. associate-*l* 37.2%

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

      6. prod-diff 18.0%

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

   3. Applied egg-rr 18.0%

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

   4. Taylor expanded in z around 0 37.2%

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

      1. distribute-rgt1-in 37.2%

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

      2. metadata-eval 37.2%

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

      3. mul0-lft 37.2%

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

   6. Simplified 37.2%

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

   7. Taylor expanded in y around 0 61.9%

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

      1. associate-*r/ 61.9%

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

      2. times-frac 99.9%

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

      3. metadata-eval 99.9%

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

      4. associate-/r* 99.9%

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

      5. neg-mul-1 99.9%

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

      6. associate-*l/ 99.9%

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

      7. *-lft-identity 99.9%

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

   9. Simplified 99.9%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 2 \cdot 10^{+269}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \end{array} \]

Alternative 2: 72.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-96} \lor \neg \left(t \leq 3.3 \cdot 10^{-17}\right) \land \left(t \leq 11500000 \lor \neg \left(t \leq 3.2 \cdot 10^{+115}\right)\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -7.8e-96)
         (and (not (<= t 3.3e-17))
              (or (<= t 11500000.0) (not (<= t 3.2e+115)))))
   (/ (/ (- x) t) z)
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -7.8e-96) || (!(t <= 3.3e-17) && ((t <= 11500000.0) || !(t <= 3.2e+115)))) {
		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 ((t <= (-7.8d-96)) .or. (.not. (t <= 3.3d-17)) .and. (t <= 11500000.0d0) .or. (.not. (t <= 3.2d+115))) 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 ((t <= -7.8e-96) || (!(t <= 3.3e-17) && ((t <= 11500000.0) || !(t <= 3.2e+115)))) {
		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 (t <= -7.8e-96) or (not (t <= 3.3e-17) and ((t <= 11500000.0) or not (t <= 3.2e+115))):
		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 ((t <= -7.8e-96) || (!(t <= 3.3e-17) && ((t <= 11500000.0) || !(t <= 3.2e+115))))
		tmp = Float64(Float64(Float64(-x) / t) / 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 ((t <= -7.8e-96) || (~((t <= 3.3e-17)) && ((t <= 11500000.0) || ~((t <= 3.2e+115)))))
		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[Or[LessEqual[t, -7.8e-96], And[N[Not[LessEqual[t, 3.3e-17]], $MachinePrecision], Or[LessEqual[t, 11500000.0], N[Not[LessEqual[t, 3.2e+115]], $MachinePrecision]]]], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -7.7999999999999997e-96 or 3.3e-17 < t < 1.15e7 or 3.2e115 < t

   1. Initial program 90.7%

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

   2. Taylor expanded in y around 0 63.2%

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

      1. mul-1-neg 63.2%

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

      2. associate-/r* 68.4%

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

      3. distribute-neg-frac 68.4%

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

   4. Simplified 68.4%

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

   if -7.7999999999999997e-96 < t < 3.3e-17 or 1.15e7 < t < 3.2e115

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 69.9%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.8 \cdot 10^{-96} \lor \neg \left(t \leq 3.3 \cdot 10^{-17}\right) \land \left(t \leq 11500000 \lor \neg \left(t \leq 3.2 \cdot 10^{+115}\right)\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 3: 71.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} t_1 := \frac{\frac{x}{z}}{-t}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+154}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -9 \cdot 10^{+90}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x z) (- t))))
   (if (<= z -6.5e+154)
     t_1
     (if (<= z -3.1e+136)
       (/ x y)
       (if (<= z -9e+90) t_1 (if (<= z 8e-81) (/ x y) (/ (/ (- x) t) z)))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double t_1 = (x / z) / -t;
	double tmp;
	if (z <= -6.5e+154) {
		tmp = t_1;
	} else if (z <= -3.1e+136) {
		tmp = x / y;
	} else if (z <= -9e+90) {
		tmp = t_1;
	} else if (z <= 8e-81) {
		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) :: t_1
    real(8) :: tmp
    t_1 = (x / z) / -t
    if (z <= (-6.5d+154)) then
        tmp = t_1
    else if (z <= (-3.1d+136)) then
        tmp = x / y
    else if (z <= (-9d+90)) then
        tmp = t_1
    else if (z <= 8d-81) 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 t_1 = (x / z) / -t;
	double tmp;
	if (z <= -6.5e+154) {
		tmp = t_1;
	} else if (z <= -3.1e+136) {
		tmp = x / y;
	} else if (z <= -9e+90) {
		tmp = t_1;
	} else if (z <= 8e-81) {
		tmp = x / y;
	} else {
		tmp = (-x / t) / z;
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	t_1 = (x / z) / -t
	tmp = 0
	if z <= -6.5e+154:
		tmp = t_1
	elif z <= -3.1e+136:
		tmp = x / y
	elif z <= -9e+90:
		tmp = t_1
	elif z <= 8e-81:
		tmp = x / y
	else:
		tmp = (-x / t) / z
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) / Float64(-t))
	tmp = 0.0
	if (z <= -6.5e+154)
		tmp = t_1;
	elseif (z <= -3.1e+136)
		tmp = Float64(x / y);
	elseif (z <= -9e+90)
		tmp = t_1;
	elseif (z <= 8e-81)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(Float64(-x) / t) / z);
	end
	return tmp
end

MATLAB

z, t = num2cell(sort([z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) / -t;
	tmp = 0.0;
	if (z <= -6.5e+154)
		tmp = t_1;
	elseif (z <= -3.1e+136)
		tmp = x / y;
	elseif (z <= -9e+90)
		tmp = t_1;
	elseif (z <= 8e-81)
		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_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] / (-t)), $MachinePrecision]}, If[LessEqual[z, -6.5e+154], t$95$1, If[LessEqual[z, -3.1e+136], N[(x / y), $MachinePrecision], If[LessEqual[z, -9e+90], t$95$1, If[LessEqual[z, 8e-81], N[(x / y), $MachinePrecision], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5000000000000005e154 or -3.09999999999999983e136 < z < -9e90

   1. Initial program 88.8%

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

      1. *-un-lft-identity 88.8%

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

      2. *-commutative 88.8%

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

      3. *-commutative 88.8%

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

      4. add-sqr-sqrt 47.5%

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

      5. associate-*l* 47.5%

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

      6. prod-diff 40.4%

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

   3. Applied egg-rr 40.4%

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

   4. Taylor expanded in z around 0 47.5%

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

      1. distribute-rgt1-in 47.5%

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

      2. metadata-eval 47.5%

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

      3. mul0-lft 47.5%

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

   6. Simplified 47.5%

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

   7. Taylor expanded in y around 0 71.0%

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

      1. associate-*r/ 71.0%

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

      2. times-frac 82.1%

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

      3. metadata-eval 82.1%

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

      4. associate-/r* 82.1%

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

      5. neg-mul-1 82.1%

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

      6. associate-*l/ 82.0%

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

      7. *-lft-identity 82.0%

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

   9. Simplified 82.0%

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

   if -6.5000000000000005e154 < z < -3.09999999999999983e136 or -9e90 < z < 7.9999999999999997e-81

   1. Initial program 99.1%

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

   2. Taylor expanded in y around inf 73.1%

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

   if 7.9999999999999997e-81 < z

   1. Initial program 92.7%

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

   2. Taylor expanded in y around 0 65.8%

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

      1. mul-1-neg 65.8%

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

      2. associate-/r* 67.4%

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

      3. distribute-neg-frac 67.4%

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

   4. Simplified 67.4%

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

4. Final simplification 72.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+154}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \leq -3.1 \cdot 10^{+136}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{elif}\;z \leq -9 \cdot 10^{+90}:\\ \;\;\;\;\frac{\frac{x}{z}}{-t}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{-81}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]

Alternative 4: 73.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{-18} \lor \neg \left(y \leq 3.3 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.26e-18) (not (<= y 3.3e-11))) (/ x y) (/ (- x) (* z t))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.26e-18) || !(y <= 3.3e-11)) {
		tmp = x / y;
	} 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 ((y <= (-1.26d-18)) .or. (.not. (y <= 3.3d-11))) then
        tmp = x / y
    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 ((y <= -1.26e-18) || !(y <= 3.3e-11)) {
		tmp = x / y;
	} else {
		tmp = -x / (z * t);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (y <= -1.26e-18) or not (y <= 3.3e-11):
		tmp = x / y
	else:
		tmp = -x / (z * t)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.26e-18) || !(y <= 3.3e-11))
		tmp = Float64(x / y);
	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 ((y <= -1.26e-18) || ~((y <= 3.3e-11)))
		tmp = x / y;
	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[Or[LessEqual[y, -1.26e-18], N[Not[LessEqual[y, 3.3e-11]], $MachinePrecision]], N[(x / y), $MachinePrecision], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.26000000000000004e-18 or 3.3000000000000002e-11 < y

   1. Initial program 95.7%

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

   2. Taylor expanded in y around inf 77.8%

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

   if -1.26000000000000004e-18 < y < 3.3000000000000002e-11

   1. Initial program 95.2%

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

   2. Taylor expanded in y around 0 71.1%

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

      1. associate-*r/ 71.1%

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

      2. neg-mul-1 71.1%

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

   4. Simplified 71.1%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.26 \cdot 10^{-18} \lor \neg \left(y \leq 3.3 \cdot 10^{-11}\right):\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 5: 58.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+235} \lor \neg \left(z \leq 3.8 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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 -4.6e+235) (not (<= z 3.8e+57))) (/ x (* z t)) (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -4.6e+235) || !(z <= 3.8e+57)) {
		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 <= (-4.6d+235)) .or. (.not. (z <= 3.8d+57))) 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 <= -4.6e+235) || !(z <= 3.8e+57)) {
		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 <= -4.6e+235) or not (z <= 3.8e+57):
		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 <= -4.6e+235) || !(z <= 3.8e+57))
		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 <= -4.6e+235) || ~((z <= 3.8e+57)))
		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, -4.6e+235], N[Not[LessEqual[z, 3.8e+57]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.6e235 or 3.7999999999999999e57 < z

   1. Initial program 87.3%

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

   2. Taylor expanded in y around 0 73.5%

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

      1. mul-1-neg 73.5%

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

      2. associate-/r* 76.3%

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

      3. distribute-neg-frac 76.3%

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

   4. Simplified 76.3%

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

      1. expm1-log1p-u 62.2%

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

      2. expm1-udef 36.1%

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

      3. add-sqr-sqrt 21.2%

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

      4. sqrt-unprod 36.2%

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

      5. sqr-neg 36.2%

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

      6. sqrt-unprod 21.9%

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

      7. add-sqr-sqrt 34.6%

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

   6. Applied egg-rr 34.6%

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

      1. expm1-def 31.3%

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

      2. expm1-log1p 31.5%

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

      3. associate-/r* 31.8%

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

   8. Simplified 31.8%

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

   if -4.6e235 < z < 3.7999999999999999e57

   1. Initial program 97.9%

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

   2. Taylor expanded in y around inf 62.8%

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

4. Final simplification 55.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+235} \lor \neg \left(z \leq 3.8 \cdot 10^{+57}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 58.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= z -9.2e+157) (/ (/ x z) t) (if (<= z 3.8e+57) (/ x y) (/ x (* z t)))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.2e+157) {
		tmp = (x / z) / t;
	} else if (z <= 3.8e+57) {
		tmp = x / y;
	} 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 <= (-9.2d+157)) then
        tmp = (x / z) / t
    else if (z <= 3.8d+57) then
        tmp = x / y
    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 <= -9.2e+157) {
		tmp = (x / z) / t;
	} else if (z <= 3.8e+57) {
		tmp = x / y;
	} else {
		tmp = x / (z * t);
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if z <= -9.2e+157:
		tmp = (x / z) / t
	elif z <= 3.8e+57:
		tmp = x / y
	else:
		tmp = x / (z * t)
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9.2e+157)
		tmp = Float64(Float64(x / z) / t);
	elseif (z <= 3.8e+57)
		tmp = Float64(x / y);
	else
		tmp = 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 <= -9.2e+157)
		tmp = (x / z) / t;
	elseif (z <= 3.8e+57)
		tmp = x / y;
	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[z, -9.2e+157], N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 3.8e+57], N[(x / y), $MachinePrecision], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.20000000000000015e157

   1. Initial program 88.3%

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

   2. Taylor expanded in y around 0 71.1%

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

      1. mul-1-neg 71.1%

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

      2. associate-/r* 71.0%

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

      3. distribute-neg-frac 71.0%

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

   4. Simplified 71.0%

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

      1. expm1-log1p-u 64.3%

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

      2. expm1-udef 40.6%

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

      3. add-sqr-sqrt 26.1%

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

      4. sqrt-unprod 40.2%

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

      5. sqr-neg 40.2%

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

      6. sqrt-unprod 23.4%

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

      7. add-sqr-sqrt 40.3%

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

   6. Applied egg-rr 40.3%

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

      1. expm1-def 35.1%

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

      2. expm1-log1p 35.1%

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

      3. associate-/r* 35.3%

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

   8. Simplified 35.3%

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

      1. expm1-log1p-u 35.3%

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

      2. expm1-udef 40.3%

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

   10. Applied egg-rr 40.3%

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

       1. expm1-def 35.3%

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

       2. expm1-log1p 35.3%

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

       3. *-commutative 35.3%

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

       4. associate-/r* 39.7%

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

   12. Simplified 39.7%

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

   if -9.20000000000000015e157 < z < 3.7999999999999999e57

   1. Initial program 98.7%

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

   2. Taylor expanded in y around inf 66.4%

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

   if 3.7999999999999999e57 < z

   1. Initial program 88.1%

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

   2. Taylor expanded in y around 0 74.0%

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

      1. mul-1-neg 74.0%

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

      2. associate-/r* 77.6%

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

      3. distribute-neg-frac 77.6%

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

   4. Simplified 77.6%

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

      1. expm1-log1p-u 60.1%

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

      2. expm1-udef 38.1%

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

      3. add-sqr-sqrt 24.1%

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

      4. sqrt-unprod 38.2%

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

      5. sqr-neg 38.2%

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

      6. sqrt-unprod 22.8%

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

      7. add-sqr-sqrt 36.2%

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

   6. Applied egg-rr 36.2%

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

      1. expm1-def 32.2%

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

      2. expm1-log1p 32.5%

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

      3. associate-/r* 32.8%

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

   8. Simplified 32.8%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+157}:\\ \;\;\;\;\frac{\frac{x}{z}}{t}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+57}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \end{array} \]

Alternative 7: 55.0% 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 95.4%

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

2. Taylor expanded in y around inf 54.8%

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

3. Final simplification 54.8%

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

Developer target: 96.6% 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 2023320 
(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))))