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

Percentage Accurate: 96.0% → 99.7%

Time: 8.4s

Alternatives: 9

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: 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: 99.7% accurate, 0.1× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ (- x) t) z)
   (if (<= (* z t) 5e+228) (/ x (fma z (- t) y)) (/ (/ x (- z)) t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 5e+228) {
		tmp = x / fma(z, -t, y);
	} else {
		tmp = (x / -z) / t;
	}
	return tmp;
}

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / t) / z);
	elseif (Float64(z * t) <= 5e+228)
		tmp = Float64(x / fma(z, Float64(-t), y));
	else
		tmp = Float64(Float64(x / Float64(-z)) / t);
	end
	return tmp
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+228], 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) < -inf.0

   1. Initial program 74.1%

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

   3. Taylor expanded in y around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. associate-/r* 99.9%

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

      3. distribute-neg-frac 99.9%

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

   5. Simplified 99.9%

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

   if -inf.0 < (*.f64 z t) < 5e228

   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%

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

   if 5e228 < (*.f64 z t)

   1. Initial program 82.5%

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

   3. Taylor expanded in y around 0 82.5%

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

      1. mul-1-neg 82.5%

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

      2. associate-/r* 100.0%

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

      3. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

      1. expm1-log1p-u 99.9%

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

      2. expm1-udef 76.5%

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

      3. add-sqr-sqrt 50.7%

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

      4. sqrt-unprod 76.5%

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

      5. sqr-neg 76.5%

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

      6. sqrt-unprod 50.9%

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

      7. add-sqr-sqrt 72.5%

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

   7. Applied egg-rr 72.5%

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

      1. expm1-def 72.0%

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

      2. expm1-log1p 72.0%

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

      3. associate-/r* 72.3%

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

   9. Simplified 72.3%

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

       1. add-sqr-sqrt 27.8%

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

       2. sqrt-unprod 75.6%

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

       3. sqr-neg 75.6%

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

       4. sqrt-unprod 54.5%

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

       5. add-sqr-sqrt 82.5%

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

       6. neg-mul-1 82.5%

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

       7. times-frac 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. frac-times 82.5%

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

       2. *-commutative 82.5%

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

       3. frac-times 99.9%

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

       4. clear-num 99.9%

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

       5. frac-times 82.5%

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

       6. *-commutative 82.5%

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

       7. *-un-lft-identity 82.5%

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

       8. div-inv 82.5%

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

       9. metadata-eval 82.5%

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

   13. Applied egg-rr 82.5%

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

       1. *-commutative 82.5%

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

       2. neg-mul-1 82.5%

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

       3. associate-/l/ 99.9%

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

   15. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(z, -t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= (* z t) (- INFINITY))
   (/ (/ (- x) t) z)
   (if (<= (* z t) 5e+228) (/ x (- y (* z t))) (/ (/ x (- z)) t))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 5e+228) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / -z) / t;
	}
	return tmp;
}

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = (-x / t) / z;
	} else if ((z * t) <= 5e+228) {
		tmp = x / (y - (z * t));
	} else {
		tmp = (x / -z) / t;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = (-x / t) / z
	elif (z * t) <= 5e+228:
		tmp = x / (y - (z * t))
	else:
		tmp = (x / -z) / t
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = Float64(Float64(Float64(-x) / t) / z);
	elseif (Float64(z * t) <= 5e+228)
		tmp = Float64(x / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x / Float64(-z)) / t);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = (-x / t) / z;
	elseif ((z * t) <= 5e+228)
		tmp = x / (y - (z * t));
	else
		tmp = (x / -z) / t;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 5e+228], 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) < -inf.0

   1. Initial program 74.1%

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

   3. Taylor expanded in y around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. associate-/r* 99.9%

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

      3. distribute-neg-frac 99.9%

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

   5. Simplified 99.9%

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

   if -inf.0 < (*.f64 z t) < 5e228

   1. Initial program 99.9%

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

   if 5e228 < (*.f64 z t)

   1. Initial program 82.5%

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

   3. Taylor expanded in y around 0 82.5%

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

      1. mul-1-neg 82.5%

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

      2. associate-/r* 100.0%

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

      3. distribute-neg-frac 100.0%

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

   5. Simplified 100.0%

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

      1. expm1-log1p-u 99.9%

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

      2. expm1-udef 76.5%

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

      3. add-sqr-sqrt 50.7%

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

      4. sqrt-unprod 76.5%

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

      5. sqr-neg 76.5%

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

      6. sqrt-unprod 50.9%

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

      7. add-sqr-sqrt 72.5%

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

   7. Applied egg-rr 72.5%

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

      1. expm1-def 72.0%

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

      2. expm1-log1p 72.0%

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

      3. associate-/r* 72.3%

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

   9. Simplified 72.3%

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

       1. add-sqr-sqrt 27.8%

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

       2. sqrt-unprod 75.6%

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

       3. sqr-neg 75.6%

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

       4. sqrt-unprod 54.5%

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

       5. add-sqr-sqrt 82.5%

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

       6. neg-mul-1 82.5%

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

       7. times-frac 99.8%

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

   11. Applied egg-rr 99.8%

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

       1. frac-times 82.5%

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

       2. *-commutative 82.5%

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

       3. frac-times 99.9%

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

       4. clear-num 99.9%

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

       5. frac-times 82.5%

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

       6. *-commutative 82.5%

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

       7. *-un-lft-identity 82.5%

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

       8. div-inv 82.5%

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

       9. metadata-eval 82.5%

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

   13. Applied egg-rr 82.5%

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

       1. *-commutative 82.5%

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

       2. neg-mul-1 82.5%

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

       3. associate-/l/ 99.9%

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

   15. Simplified 99.9%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{+228}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 73.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;t \leq 27000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.5e-77)
   (/ (/ x (- z)) t)
   (if (<= t 27000.0) (/ x y) (/ -1.0 (* z (/ t x))))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.5e-77) {
		tmp = (x / -z) / t;
	} else if (t <= 27000.0) {
		tmp = x / y;
	} else {
		tmp = -1.0 / (z * (t / x));
	}
	return tmp;
}

Fortran

NOTE: x, y, 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 <= (-8.5d-77)) then
        tmp = (x / -z) / t
    else if (t <= 27000.0d0) then
        tmp = x / y
    else
        tmp = (-1.0d0) / (z * (t / x))
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.5e-77) {
		tmp = (x / -z) / t;
	} else if (t <= 27000.0) {
		tmp = x / y;
	} else {
		tmp = -1.0 / (z * (t / x));
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -8.5e-77:
		tmp = (x / -z) / t
	elif t <= 27000.0:
		tmp = x / y
	else:
		tmp = -1.0 / (z * (t / x))
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.5e-77)
		tmp = Float64(Float64(x / Float64(-z)) / t);
	elseif (t <= 27000.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(-1.0 / Float64(z * Float64(t / x)));
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -8.5e-77)
		tmp = (x / -z) / t;
	elseif (t <= 27000.0)
		tmp = x / y;
	else
		tmp = -1.0 / (z * (t / x));
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -8.5e-77], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 27000.0], N[(x / y), $MachinePrecision], N[(-1.0 / N[(z * N[(t / x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.4999999999999998e-77

   1. Initial program 93.5%

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

   3. Taylor expanded in y around 0 59.6%

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

      1. mul-1-neg 59.6%

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

      2. associate-/r* 66.1%

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

      3. distribute-neg-frac 66.1%

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

   5. Simplified 66.1%

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

      1. expm1-log1p-u 57.7%

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

      2. expm1-udef 38.2%

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

      3. add-sqr-sqrt 25.3%

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

      4. sqrt-unprod 30.1%

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

      5. sqr-neg 30.1%

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

      6. sqrt-unprod 21.0%

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

      7. add-sqr-sqrt 28.8%

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

   7. Applied egg-rr 28.8%

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

      1. expm1-def 25.9%

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

      2. expm1-log1p 27.6%

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

      3. associate-/r* 25.5%

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

   9. Simplified 25.5%

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

       1. add-sqr-sqrt 10.4%

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

       2. sqrt-unprod 40.0%

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

       3. sqr-neg 40.0%

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

       4. sqrt-unprod 32.7%

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

       5. add-sqr-sqrt 59.6%

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

       6. neg-mul-1 59.6%

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

       7. times-frac 62.4%

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

   11. Applied egg-rr 62.4%

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

       1. frac-times 59.6%

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

       2. *-commutative 59.6%

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

       3. frac-times 66.1%

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

       4. clear-num 66.1%

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

       5. frac-times 59.6%

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

       6. *-commutative 59.6%

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

       7. *-un-lft-identity 59.6%

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

       8. div-inv 59.6%

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

       9. metadata-eval 59.6%

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

   13. Applied egg-rr 59.6%

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

       1. *-commutative 59.6%

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

       2. neg-mul-1 59.6%

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

       3. associate-/l/ 62.5%

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

   15. Simplified 62.5%

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

   if -8.4999999999999998e-77 < t < 27000

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.2%

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

   if 27000 < t

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0 76.9%

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

      1. mul-1-neg 76.9%

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

      2. associate-/r* 81.4%

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

      3. distribute-neg-frac 81.4%

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

   5. Simplified 81.4%

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

      1. expm1-log1p-u 70.5%

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

      2. expm1-udef 50.7%

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

      3. add-sqr-sqrt 35.3%

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

      4. sqrt-unprod 45.2%

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

      5. sqr-neg 45.2%

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

      6. sqrt-unprod 28.4%

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

      7. add-sqr-sqrt 43.7%

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

   7. Applied egg-rr 43.7%

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

      1. expm1-def 41.9%

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

      2. expm1-log1p 42.0%

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

      3. associate-/r* 40.5%

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

   9. Simplified 40.5%

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

       1. add-sqr-sqrt 16.0%

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

       2. sqrt-unprod 51.5%

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

       3. sqr-neg 51.5%

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

       4. sqrt-unprod 43.3%

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

       5. add-sqr-sqrt 76.9%

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

       6. neg-mul-1 76.9%

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

       7. times-frac 73.8%

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

   11. Applied egg-rr 73.8%

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

       1. frac-times 76.9%

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

       2. *-commutative 76.9%

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

       3. frac-times 81.4%

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

       4. clear-num 82.8%

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

       5. frac-times 82.0%

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

       6. metadata-eval 82.0%

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

   13. Applied egg-rr 82.0%

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

4. Final simplification 73.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;t \leq 27000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-1}{z \cdot \frac{t}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 70.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-77} \lor \neg \left(t \leq 27500\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -5.2e-77) (not (<= t 27500.0))) (/ (- x) (* z t)) (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.2e-77) || !(t <= 27500.0)) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, 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 <= (-5.2d-77)) .or. (.not. (t <= 27500.0d0))) then
        tmp = -x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -5.2e-77) || !(t <= 27500.0)) {
		tmp = -x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (t <= -5.2e-77) or not (t <= 27500.0):
		tmp = -x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -5.2e-77) || !(t <= 27500.0))
		tmp = Float64(Float64(-x) / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -5.2e-77) || ~((t <= 27500.0)))
		tmp = -x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -5.2e-77], N[Not[LessEqual[t, 27500.0]], $MachinePrecision]], N[((-x) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2000000000000002e-77 or 27500 < t

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0 67.4%

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

      1. associate-*r/ 67.4%

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

      2. neg-mul-1 67.4%

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

   5. Simplified 67.4%

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

   if -5.2000000000000002e-77 < t < 27500

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.2%

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

4. Final simplification 71.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.2 \cdot 10^{-77} \lor \neg \left(t \leq 27500\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-77} \lor \neg \left(t \leq 27500\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -4.3e-77) (not (<= t 27500.0))) (/ (/ (- x) t) z) (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.3e-77) || !(t <= 27500.0)) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, 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 <= (-4.3d-77)) .or. (.not. (t <= 27500.0d0))) then
        tmp = (-x / t) / z
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -4.3e-77) || !(t <= 27500.0)) {
		tmp = (-x / t) / z;
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (t <= -4.3e-77) or not (t <= 27500.0):
		tmp = (-x / t) / z
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -4.3e-77) || !(t <= 27500.0))
		tmp = Float64(Float64(Float64(-x) / t) / z);
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -4.3e-77) || ~((t <= 27500.0)))
		tmp = (-x / t) / z;
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -4.3e-77], N[Not[LessEqual[t, 27500.0]], $MachinePrecision]], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -4.3000000000000002e-77 or 27500 < t

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0 67.4%

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

      1. mul-1-neg 67.4%

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

      2. associate-/r* 73.1%

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

      3. distribute-neg-frac 73.1%

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

   5. Simplified 73.1%

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

   if -4.3000000000000002e-77 < t < 27500

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.2%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{-77} \lor \neg \left(t \leq 27500\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;t \leq 27000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -9.5e-77)
   (/ (/ x (- z)) t)
   (if (<= t 27000.0) (/ x y) (/ (/ (- x) t) z))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9.5e-77) {
		tmp = (x / -z) / t;
	} else if (t <= 27000.0) {
		tmp = x / y;
	} else {
		tmp = (-x / t) / z;
	}
	return tmp;
}

Fortran

NOTE: x, y, 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 <= (-9.5d-77)) then
        tmp = (x / -z) / t
    else if (t <= 27000.0d0) then
        tmp = x / y
    else
        tmp = (-x / t) / z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -9.5e-77) {
		tmp = (x / -z) / t;
	} else if (t <= 27000.0) {
		tmp = x / y;
	} else {
		tmp = (-x / t) / z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -9.5e-77:
		tmp = (x / -z) / t
	elif t <= 27000.0:
		tmp = x / y
	else:
		tmp = (-x / t) / z
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -9.5e-77)
		tmp = Float64(Float64(x / Float64(-z)) / t);
	elseif (t <= 27000.0)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(Float64(-x) / t) / z);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -9.5e-77)
		tmp = (x / -z) / t;
	elseif (t <= 27000.0)
		tmp = x / y;
	else
		tmp = (-x / t) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -9.5e-77], N[(N[(x / (-z)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t, 27000.0], N[(x / y), $MachinePrecision], N[(N[((-x) / t), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.5000000000000005e-77

   1. Initial program 93.5%

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

   3. Taylor expanded in y around 0 59.6%

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

      1. mul-1-neg 59.6%

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

      2. associate-/r* 66.1%

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

      3. distribute-neg-frac 66.1%

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

   5. Simplified 66.1%

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

      1. expm1-log1p-u 57.7%

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

      2. expm1-udef 38.2%

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

      3. add-sqr-sqrt 25.3%

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

      4. sqrt-unprod 30.1%

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

      5. sqr-neg 30.1%

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

      6. sqrt-unprod 21.0%

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

      7. add-sqr-sqrt 28.8%

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

   7. Applied egg-rr 28.8%

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

      1. expm1-def 25.9%

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

      2. expm1-log1p 27.6%

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

      3. associate-/r* 25.5%

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

   9. Simplified 25.5%

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

       1. add-sqr-sqrt 10.4%

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

       2. sqrt-unprod 40.0%

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

       3. sqr-neg 40.0%

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

       4. sqrt-unprod 32.7%

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

       5. add-sqr-sqrt 59.6%

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

       6. neg-mul-1 59.6%

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

       7. times-frac 62.4%

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

   11. Applied egg-rr 62.4%

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

       1. frac-times 59.6%

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

       2. *-commutative 59.6%

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

       3. frac-times 66.1%

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

       4. clear-num 66.1%

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

       5. frac-times 59.6%

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

       6. *-commutative 59.6%

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

       7. *-un-lft-identity 59.6%

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

       8. div-inv 59.6%

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

       9. metadata-eval 59.6%

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

   13. Applied egg-rr 59.6%

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

       1. *-commutative 59.6%

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

       2. neg-mul-1 59.6%

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

       3. associate-/l/ 62.5%

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

   15. Simplified 62.5%

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

   if -9.5000000000000005e-77 < t < 27000

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 76.2%

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

   if 27000 < t

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0 76.9%

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

      1. mul-1-neg 76.9%

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

      2. associate-/r* 81.4%

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

      3. distribute-neg-frac 81.4%

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

   5. Simplified 81.4%

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

4. Final simplification 73.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{-77}:\\ \;\;\;\;\frac{\frac{x}{-z}}{t}\\ \mathbf{elif}\;t \leq 27000:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 58.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+36} \lor \neg \left(t \leq 2.8 \cdot 10^{+174}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.25e+36) (not (<= t 2.8e+174))) (/ x (* z t)) (/ x y)))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.25e+36) || !(t <= 2.8e+174)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Fortran

NOTE: x, y, 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 <= (-1.25d+36)) .or. (.not. (t <= 2.8d+174))) then
        tmp = x / (z * t)
    else
        tmp = x / y
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.25e+36) || !(t <= 2.8e+174)) {
		tmp = x / (z * t);
	} else {
		tmp = x / y;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if (t <= -1.25e+36) or not (t <= 2.8e+174):
		tmp = x / (z * t)
	else:
		tmp = x / y
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.25e+36) || !(t <= 2.8e+174))
		tmp = Float64(x / Float64(z * t));
	else
		tmp = Float64(x / y);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.25e+36) || ~((t <= 2.8e+174)))
		tmp = x / (z * t);
	else
		tmp = x / y;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.25e+36], N[Not[LessEqual[t, 2.8e+174]], $MachinePrecision]], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x / y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.24999999999999994e36 or 2.7999999999999999e174 < t

   1. Initial program 90.5%

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

   3. Taylor expanded in y around 0 68.1%

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

      1. mul-1-neg 68.1%

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

      2. associate-/r* 76.9%

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

      3. distribute-neg-frac 76.9%

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

   5. Simplified 76.9%

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

      1. expm1-log1p-u 68.8%

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

      2. expm1-udef 47.2%

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

      3. add-sqr-sqrt 36.0%

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

      4. sqrt-unprod 43.4%

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

      5. sqr-neg 43.4%

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

      6. sqrt-unprod 30.5%

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

      7. add-sqr-sqrt 42.4%

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

   7. Applied egg-rr 42.4%

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

      1. expm1-def 40.0%

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

      2. expm1-log1p 41.3%

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

      3. associate-/r* 38.4%

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

   9. Simplified 38.4%

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

   if -1.24999999999999994e36 < t < 2.7999999999999999e174

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 64.5%

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

4. Final simplification 55.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+36} \lor \neg \left(t \leq 2.8 \cdot 10^{+174}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 59.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+175}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.5e+35)
   (/ x (* z t))
   (if (<= t 2.05e+175) (/ x y) (/ (/ x t) z))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.5e+35) {
		tmp = x / (z * t);
	} else if (t <= 2.05e+175) {
		tmp = x / y;
	} else {
		tmp = (x / t) / z;
	}
	return tmp;
}

Fortran

NOTE: x, y, 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 <= (-7.5d+35)) then
        tmp = x / (z * t)
    else if (t <= 2.05d+175) then
        tmp = x / y
    else
        tmp = (x / t) / z
    end if
    code = tmp
end function

Java

assert x < y && y < z && z < t;
public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.5e+35) {
		tmp = x / (z * t);
	} else if (t <= 2.05e+175) {
		tmp = x / y;
	} else {
		tmp = (x / t) / z;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	tmp = 0
	if t <= -7.5e+35:
		tmp = x / (z * t)
	elif t <= 2.05e+175:
		tmp = x / y
	else:
		tmp = (x / t) / z
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.5e+35)
		tmp = Float64(x / Float64(z * t));
	elseif (t <= 2.05e+175)
		tmp = Float64(x / y);
	else
		tmp = Float64(Float64(x / t) / z);
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -7.5e+35)
		tmp = x / (z * t);
	elseif (t <= 2.05e+175)
		tmp = x / y;
	else
		tmp = (x / t) / z;
	end
	tmp_2 = tmp;
end

Wolfram

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
code[x_, y_, z_, t_] := If[LessEqual[t, -7.5e+35], N[(x / N[(z * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.05e+175], N[(x / y), $MachinePrecision], N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -7.4999999999999999e35

   1. Initial program 91.0%

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

   3. Taylor expanded in y around 0 63.2%

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

      1. mul-1-neg 63.2%

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

      2. associate-/r* 72.4%

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

      3. distribute-neg-frac 72.4%

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

   5. Simplified 72.4%

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

      1. expm1-log1p-u 64.9%

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

      2. expm1-udef 42.5%

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

      3. add-sqr-sqrt 29.7%

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

      4. sqrt-unprod 39.5%

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

      5. sqr-neg 39.5%

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

      6. sqrt-unprod 27.6%

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

      7. add-sqr-sqrt 37.8%

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

   7. Applied egg-rr 37.8%

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

      1. expm1-def 34.1%

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

      2. expm1-log1p 36.0%

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

      3. associate-/r* 33.1%

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

   9. Simplified 33.1%

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

   if -7.4999999999999999e35 < t < 2.04999999999999989e175

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 64.5%

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

   if 2.04999999999999989e175 < t

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 0 75.9%

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

      1. mul-1-neg 75.9%

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

      2. associate-/r* 84.1%

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

      3. distribute-neg-frac 84.1%

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

   5. Simplified 84.1%

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

      1. expm1-log1p-u 75.1%

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

      2. expm1-udef 54.9%

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

      3. add-sqr-sqrt 46.2%

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

      4. sqrt-unprod 49.8%

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

      5. sqr-neg 49.8%

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

      6. sqrt-unprod 35.2%

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

      7. add-sqr-sqrt 49.9%

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

   7. Applied egg-rr 49.9%

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

      1. expm1-def 49.6%

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

      2. expm1-log1p 49.7%

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

      3. associate-/r* 47.0%

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

   9. Simplified 47.0%

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

       1. add-sqr-sqrt 17.6%

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

       2. sqrt-unprod 54.5%

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

       3. sqr-neg 54.5%

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

       4. sqrt-unprod 44.3%

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

       5. add-sqr-sqrt 75.9%

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

       6. neg-mul-1 75.9%

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

       7. times-frac 75.3%

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

   11. Applied egg-rr 75.3%

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

       1. *-commutative 75.3%

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

       2. associate-*l/ 84.1%

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

       3. frac-2neg 84.1%

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

       4. metadata-eval 84.1%

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

       5. div-inv 84.1%

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

       6. add-sqr-sqrt 0.0%

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

       7. sqrt-unprod 50.3%

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

       8. sqr-neg 50.3%

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

       9. sqrt-unprod 49.7%

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

       10. add-sqr-sqrt 49.7%

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

   13. Applied egg-rr 49.7%

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

4. Final simplification 55.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{+35}:\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{elif}\;t \leq 2.05 \cdot 10^{+175}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 54.1% accurate, 2.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.4%

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

3. Taylor expanded in y around inf 53.1%

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

4. Final simplification 53.1%

   \[\leadsto \frac{x}{y} \]
5. Add Preprocessing

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 2024010 
(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))))