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

Percentage Accurate: 95.9% → 99.6%

Time: 7.3s

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: 95.9% 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: 99.6% accurate, 0.1× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \end{array} \]

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) -1e+270)
   (/ (/ x t) (- z))
   (if (<= (* z t) 2e+220) (/ 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) <= -1e+270) {
		tmp = (x / t) / -z;
	} else if ((z * t) <= 2e+220) {
		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) <= -1e+270)
		tmp = Float64(Float64(x / t) / Float64(-z));
	elseif (Float64(z * t) <= 2e+220)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(Float64(-x) / 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], -1e+270], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+220], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1e270

   1. Initial program 75.4%

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

   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%

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

      1. neg-mul-1 75.4%

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

      2. *-commutative 75.4%

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

      3. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. frac-2neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. un-div-inv 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -1e270 < (*.f64 z t) < 2e220

   1. Initial program 99.8%

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

      1. sub-neg 99.8%

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

      2. +-commutative 99.8%

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

      3. distribute-lft-neg-in 99.8%

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

      4. fma-def 99.8%

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

   3. Applied egg-rr 99.8%

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

   if 2e220 < (*.f64 z t)

   1. Initial program 76.2%

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

      1. clear-num 76.2%

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

      2. associate-/r/ 76.2%

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

   3. Applied egg-rr 76.2%

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

   4. Taylor expanded in y around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. neg-mul-1 76.2%

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

      3. *-commutative 76.2%

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

      4. associate-/r* 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 2: 78.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 500000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{z}{\frac{x}{t}}}\\ \end{array} \end{array} \]

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+26)
   (/ (/ x t) (- z))
   (if (<= (* z t) 500000.0) (/ x y) (/ -1.0 (/ z (/ x t))))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+26) {
		tmp = (x / t) / -z;
	} else if ((z * t) <= 500000.0) {
		tmp = x / y;
	} else {
		tmp = -1.0 / (z / (x / 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+26)) then
        tmp = (x / t) / -z
    else if ((z * t) <= 500000.0d0) then
        tmp = x / y
    else
        tmp = (-1.0d0) / (z / (x / 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+26) {
		tmp = (x / t) / -z;
	} else if ((z * t) <= 500000.0) {
		tmp = x / y;
	} else {
		tmp = -1.0 / (z / (x / t));
	}
	return tmp;
}

Python

[z, t] = sort([z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -2e+26:
		tmp = (x / t) / -z
	elif (z * t) <= 500000.0:
		tmp = x / y
	else:
		tmp = -1.0 / (z / (x / t))
	return tmp

Julia

z, t = sort([z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= -2e+26)
		tmp = Float64(Float64(x / t) / Float64(-z));
	elseif (Float64(z * t) <= 500000.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(-1.0 / Float64(z / Float64(x / 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+26)
		tmp = (x / t) / -z;
	elseif ((z * t) <= 500000.0)
		tmp = x / y;
	else
		tmp = -1.0 / (z / (x / 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+26], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 500000.0], N[(x / y), $MachinePrecision], N[(-1.0 / N[(z / N[(x / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -2.0000000000000001e26

   1. Initial program 91.5%

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

   2. Taylor expanded in y around 0 72.5%

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

      1. associate-*r/ 72.5%

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

      2. neg-mul-1 72.5%

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

   4. Simplified 72.5%

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

      1. neg-mul-1 72.5%

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

      2. *-commutative 72.5%

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

      3. times-frac 78.2%

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

   6. Applied egg-rr 78.2%

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

      1. *-commutative 78.2%

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

      2. frac-2neg 78.2%

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

      3. metadata-eval 78.2%

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

      4. un-div-inv 78.3%

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

   8. Applied egg-rr 78.3%

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

   if -2.0000000000000001e26 < (*.f64 z t) < 5e5

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 85.3%

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

   if 5e5 < (*.f64 z t)

   1. Initial program 92.4%

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

      1. add-cube-cbrt 91.4%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x}{y - z \cdot t}} \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}\right) \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}} \]

      2. pow3 91.4%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]

   3. Applied egg-rr 91.4%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]

   4. Taylor expanded in y around 0 79.1%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-1 \cdot \frac{x}{t \cdot z}}}\right)}^{3} \]
   5. Step-by-step derivation

      1. associate-*r/ 79.9%

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

      2. neg-mul-1 79.9%

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

   6. Simplified 79.1%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{-x}{t \cdot z}}}\right)}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 79.9%

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

      2. neg-mul-1 79.9%

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

      3. associate-/l* 80.8%

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

      4. *-commutative 80.8%

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

      5. add-sqr-sqrt 46.1%

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

      6. sqrt-unprod 49.5%

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

      7. sqr-neg 49.5%

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

      8. sqrt-unprod 14.1%

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

      9. add-sqr-sqrt 31.3%

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

      10. associate-/l* 32.5%

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

      11. add-sqr-sqrt 15.2%

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

      12. sqrt-unprod 49.5%

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

      13. sqr-neg 49.5%

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

      14. sqrt-unprod 46.1%

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

      15. add-sqr-sqrt 81.8%

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

   8. Applied egg-rr 81.8%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+26}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 500000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{\frac{z}{\frac{x}{t}}}\\ \end{array} \]

Alternative 3: 99.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \end{array} \]

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) -1e+270)
   (/ (/ x t) (- z))
   (if (<= (* z t) 2e+220) (/ 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) <= -1e+270) {
		tmp = (x / t) / -z;
	} else if ((z * t) <= 2e+220) {
		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) <= (-1d+270)) then
        tmp = (x / t) / -z
    else if ((z * t) <= 2d+220) 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) <= -1e+270) {
		tmp = (x / t) / -z;
	} else if ((z * t) <= 2e+220) {
		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) <= -1e+270:
		tmp = (x / t) / -z
	elif (z * t) <= 2e+220:
		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) <= -1e+270)
		tmp = Float64(Float64(x / t) / Float64(-z));
	elseif (Float64(z * t) <= 2e+220)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(Float64(-x) / 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) <= -1e+270)
		tmp = (x / t) / -z;
	elseif ((z * t) <= 2e+220)
		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], -1e+270], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+220], N[(x / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x) / z), $MachinePrecision] / t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1e270

   1. Initial program 75.4%

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

   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%

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

      1. neg-mul-1 75.4%

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

      2. *-commutative 75.4%

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

      3. times-frac 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. *-commutative 99.9%

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

      2. frac-2neg 99.9%

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

      3. metadata-eval 99.9%

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

      4. un-div-inv 99.9%

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

   8. Applied egg-rr 99.9%

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

   if -1e270 < (*.f64 z t) < 2e220

   1. Initial program 99.8%

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

   if 2e220 < (*.f64 z t)

   1. Initial program 76.2%

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

      1. clear-num 76.2%

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

      2. associate-/r/ 76.2%

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

   3. Applied egg-rr 76.2%

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

   4. Taylor expanded in y around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. neg-mul-1 76.2%

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

      3. *-commutative 76.2%

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

      4. associate-/r* 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 99.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -1 \cdot 10^{+270}:\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+220}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t}\\ \end{array} \]

Alternative 4: 76.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+26} \lor \neg \left(z \cdot t \leq 500000\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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+26) (not (<= (* z t) 500000.0)))
   (/ (- 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+26) || !((z * t) <= 500000.0)) {
		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+26)) .or. (.not. ((z * t) <= 500000.0d0))) 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+26) || !((z * t) <= 500000.0)) {
		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+26) or not ((z * t) <= 500000.0):
		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+26) || !(Float64(z * t) <= 500000.0))
		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+26) || ~(((z * t) <= 500000.0)))
		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+26], N[Not[LessEqual[N[(z * t), $MachinePrecision], 500000.0]], $MachinePrecision]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e26 or 5e5 < (*.f64 z t)

   1. Initial program 92.0%

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

   2. Taylor expanded in y around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. neg-mul-1 76.2%

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

   4. Simplified 76.2%

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

   if -2.0000000000000001e26 < (*.f64 z t) < 5e5

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 85.3%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+26} \lor \neg \left(z \cdot t \leq 500000\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 5: 78.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+26} \lor \neg \left(z \cdot t \leq 500000\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= (* z t) -2e+26) (not (<= (* z t) 500000.0)))
   (/ (/ x t) (- z))
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+26) || !((z * t) <= 500000.0)) {
		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 (((z * t) <= (-2d+26)) .or. (.not. ((z * t) <= 500000.0d0))) 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 (((z * t) <= -2e+26) || !((z * t) <= 500000.0)) {
		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 ((z * t) <= -2e+26) or not ((z * t) <= 500000.0):
		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 ((Float64(z * t) <= -2e+26) || !(Float64(z * t) <= 500000.0))
		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 (((z * t) <= -2e+26) || ~(((z * t) <= 500000.0)))
		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[N[(z * t), $MachinePrecision], -2e+26], N[Not[LessEqual[N[(z * t), $MachinePrecision], 500000.0]], $MachinePrecision]], N[(N[(x / t), $MachinePrecision] / (-z)), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.0000000000000001e26 or 5e5 < (*.f64 z t)

   1. Initial program 92.0%

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

   2. Taylor expanded in y around 0 76.2%

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

      1. associate-*r/ 76.2%

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

      2. neg-mul-1 76.2%

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

   4. Simplified 76.2%

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

      1. neg-mul-1 76.2%

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

      2. *-commutative 76.2%

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

      3. times-frac 80.1%

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

   6. Applied egg-rr 80.1%

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

      1. *-commutative 80.1%

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

      2. frac-2neg 80.1%

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

      3. metadata-eval 80.1%

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

      4. un-div-inv 80.2%

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

   8. Applied egg-rr 80.2%

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

   if -2.0000000000000001e26 < (*.f64 z t) < 5e5

   1. Initial program 99.9%

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

   2. Taylor expanded in y around inf 85.3%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+26} \lor \neg \left(z \cdot t \leq 500000\right):\\ \;\;\;\;\frac{\frac{x}{t}}{-z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 6: 63.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2.8 \cdot 10^{+118} \lor \neg \left(z \cdot t \leq 5.4 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

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) -2.8e+118) (not (<= (* z t) 5.4e+138)))
   (/ x (* z t))
   (/ x y)))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2.8e+118) || !((z * t) <= 5.4e+138)) {
		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) <= (-2.8d+118)) .or. (.not. ((z * t) <= 5.4d+138))) 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) <= -2.8e+118) || !((z * t) <= 5.4e+138)) {
		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) <= -2.8e+118) or not ((z * t) <= 5.4e+138):
		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) <= -2.8e+118) || !(Float64(z * t) <= 5.4e+138))
		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) <= -2.8e+118) || ~(((z * t) <= 5.4e+138)))
		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], -2.8e+118], N[Not[LessEqual[N[(z * t), $MachinePrecision], 5.4e+138]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2.79999999999999986e118 or 5.40000000000000018e138 < (*.f64 z t)

   1. Initial program 86.9%

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

      1. add-cube-cbrt 86.2%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x}{y - z \cdot t}} \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}\right) \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}} \]

      2. pow3 86.2%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]

   3. Applied egg-rr 86.2%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]

   4. Taylor expanded in y around 0 81.8%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-1 \cdot \frac{x}{t \cdot z}}}\right)}^{3} \]
   5. Step-by-step derivation

      1. associate-*r/ 82.4%

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

      2. neg-mul-1 82.4%

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

   6. Simplified 81.8%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{-x}{t \cdot z}}}\right)}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 82.4%

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

      2. add-sqr-sqrt 31.3%

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

      3. sqrt-unprod 49.0%

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

      4. sqr-neg 49.0%

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

      5. sqrt-unprod 22.1%

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

      6. add-sqr-sqrt 43.6%

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

      7. *-commutative 43.6%

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

   8. Applied egg-rr 43.6%

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

   if -2.79999999999999986e118 < (*.f64 z t) < 5.40000000000000018e138

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 71.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2.8 \cdot 10^{+118} \lor \neg \left(z \cdot t \leq 5.4 \cdot 10^{+138}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]

Alternative 7: 63.2% accurate, 0.5× speedup

Math

\[\begin{array}{l} [z, t] = \mathsf{sort}([z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+118}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: z and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) -2e+118)
   (/ x (* z t))
   (if (<= (* z t) 2e+131) (/ x y) (/ (/ x t) z))))

C

assert(z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -2e+118) {
		tmp = x / (z * t);
	} else if ((z * t) <= 2e+131) {
		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 ((z * t) <= (-2d+118)) then
        tmp = x / (z * t)
    else if ((z * t) <= 2d+131) 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 ((z * t) <= -2e+118) {
		tmp = x / (z * t);
	} else if ((z * t) <= 2e+131) {
		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 (z * t) <= -2e+118:
		tmp = x / (z * t)
	elif (z * t) <= 2e+131:
		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 (Float64(z * t) <= -2e+118)
		tmp = Float64(x / Float64(z * t));
	elseif (Float64(z * t) <= 2e+131)
		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 ((z * t) <= -2e+118)
		tmp = x / (z * t);
	elseif ((z * t) <= 2e+131)
		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[N[(z * t), $MachinePrecision], -2e+118], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+131], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (*.f64 z t) < -1.99999999999999993e118

   1. Initial program 86.4%

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

      1. add-cube-cbrt 85.5%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x}{y - z \cdot t}} \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}\right) \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}} \]

      2. pow3 85.6%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]

   3. Applied egg-rr 85.6%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]

   4. Taylor expanded in y around 0 78.6%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-1 \cdot \frac{x}{t \cdot z}}}\right)}^{3} \]
   5. Step-by-step derivation

      1. associate-*r/ 79.2%

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

      2. neg-mul-1 79.2%

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

   6. Simplified 78.6%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{-x}{t \cdot z}}}\right)}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 79.2%

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

      2. add-sqr-sqrt 35.6%

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

      3. sqrt-unprod 49.4%

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

      4. sqr-neg 49.4%

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

      5. sqrt-unprod 17.0%

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

      6. add-sqr-sqrt 40.4%

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

      7. *-commutative 40.4%

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

   8. Applied egg-rr 40.4%

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

   if -1.99999999999999993e118 < (*.f64 z t) < 1.9999999999999998e131

   1. Initial program 99.8%

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

   2. Taylor expanded in y around inf 71.4%

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

   if 1.9999999999999998e131 < (*.f64 z t)

   1. Initial program 88.3%

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

      1. add-cube-cbrt 87.7%

         \[\leadsto \color{blue}{\left(\sqrt[3]{\frac{x}{y - z \cdot t}} \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}\right) \cdot \sqrt[3]{\frac{x}{y - z \cdot t}}} \]

      2. pow3 87.7%

         \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]

   3. Applied egg-rr 87.7%

      \[\leadsto \color{blue}{{\left(\sqrt[3]{\frac{x}{y - z \cdot t}}\right)}^{3}} \]

   4. Taylor expanded in y around 0 83.8%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{-1 \cdot \frac{x}{t \cdot z}}}\right)}^{3} \]
   5. Step-by-step derivation

      1. associate-*r/ 84.4%

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

      2. neg-mul-1 84.4%

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

   6. Simplified 83.8%

      \[\leadsto {\left(\sqrt[3]{\color{blue}{\frac{-x}{t \cdot z}}}\right)}^{3} \]
   7. Step-by-step derivation

      1. rem-cube-cbrt 84.4%

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

      2. add-sqr-sqrt 27.8%

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

      3. sqrt-unprod 49.0%

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

      4. sqr-neg 49.0%

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

      5. sqrt-unprod 26.9%

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

      6. add-sqr-sqrt 45.4%

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

      7. associate-/r* 47.1%

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

   8. Applied egg-rr 47.1%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+118}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+131}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]

Alternative 8: 54.1% 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 95.9%

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

2. Taylor expanded in y around inf 54.9%

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

3. Final simplification 54.9%

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

Developer target: 96.5% 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 2023200 
(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))))