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

Percentage Accurate: 95.6% → 99.4%

Time: 9.9s

Alternatives: 10

Speedup: 0.3×

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: 95.6% 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: 99.4% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\left|\frac{\frac{x_m}{z}}{t}\right|\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\frac{x_m}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{z} \cdot \frac{\sqrt[3]{-1}}{t}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* z t) (- INFINITY))
    (fabs (/ (/ x_m z) t))
    (if (<= (* z t) 2e+196)
      (/ x_m (- y (* z t)))
      (* (/ x_m z) (/ (cbrt -1.0) t))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = fabs(((x_m / z) / t));
	} else if ((z * t) <= 2e+196) {
		tmp = x_m / (y - (z * t));
	} else {
		tmp = (x_m / z) * (cbrt(-1.0) / t);
	}
	return x_s * tmp;
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.abs(((x_m / z) / t));
	} else if ((z * t) <= 2e+196) {
		tmp = x_m / (y - (z * t));
	} else {
		tmp = (x_m / z) * (Math.cbrt(-1.0) / t);
	}
	return x_s * tmp;
}

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = abs(Float64(Float64(x_m / z) / t));
	elseif (Float64(z * t) <= 2e+196)
		tmp = Float64(x_m / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(x_m / z) * Float64(cbrt(-1.0) / t));
	end
	return Float64(x_s * tmp)
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Abs[N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+196], N[(x$95$m / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * N[(N[Power[-1.0, 1/3], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 56.1%

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

   3. Taylor expanded in y around 0 56.1%

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

      1. associate-*r/ 56.1%

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

      2. neg-mul-1 56.1%

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

   5. Simplified 56.1%

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

      1. associate-/r* 100.0%

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

      2. add-sqr-sqrt 73.2%

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

      3. sqrt-unprod 68.8%

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

      4. clear-num 68.7%

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

      5. clear-num 68.6%

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

      6. unpow-1 68.6%

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

      7. unpow-1 68.6%

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

      8. pow-prod-up 68.7%

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

      9. div-inv 68.7%

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

      10. clear-num 68.7%

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

      11. add-sqr-sqrt 15.0%

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

      12. sqrt-unprod 48.8%

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

      13. sqr-neg 48.8%

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

      14. sqrt-unprod 53.5%

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

      15. add-sqr-sqrt 68.7%

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

      16. metadata-eval 68.7%

          \[\leadsto \sqrt{{\left(z \cdot \frac{t}{x}\right)}^{\color{blue}{-2}}} \]

   7. Applied egg-rr 68.7%

      \[\leadsto \color{blue}{\sqrt{{\left(z \cdot \frac{t}{x}\right)}^{-2}}} \]
   8. Step-by-step derivation

      1. metadata-eval 68.7%

         \[\leadsto \sqrt{{\left(z \cdot \frac{t}{x}\right)}^{\color{blue}{\left(2 \cdot -1\right)}}} \]

      2. pow-sqr 68.6%

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

      3. rem-sqrt-square 74.4%

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

      4. unpow-1 74.4%

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

      5. *-commutative 74.4%

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

      6. associate-*l/ 56.1%

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

      7. associate-/l* 74.4%

         \[\leadsto \left|\frac{1}{\color{blue}{\frac{t}{\frac{x}{z}}}}\right| \]

      8. associate-/l* 74.7%

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

      9. *-lft-identity 74.7%

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

   9. Simplified 74.7%

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

   if -inf.0 < (*.f64 z t) < 1.9999999999999999e196

   1. Initial program 99.9%

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

   if 1.9999999999999999e196 < (*.f64 z t)

   1. Initial program 78.9%

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

   3. Taylor expanded in y around 0 78.9%

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

      1. mul-1-neg 78.9%

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

      2. associate-/r* 99.8%

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

      3. distribute-neg-frac 99.8%

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

   5. Simplified 99.8%

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

      1. add-cbrt-cube 73.4%

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

      2. pow3 73.4%

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

      3. distribute-neg-frac 73.4%

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

   7. Applied egg-rr 73.4%

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

   8. Taylor expanded in x around 0 78.9%

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

      1. *-commutative 78.9%

         \[\leadsto \frac{x \cdot \sqrt[3]{-1}}{\color{blue}{z \cdot t}} \]

      2. times-frac 99.8%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sqrt[3]{-1}}{t}} \]

   10. Simplified 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\left|\frac{\frac{x}{z}}{t}\right|\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \frac{\sqrt[3]{-1}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 99.5% accurate, 0.1× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\left|\frac{\frac{x_m}{z}}{t}\right|\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\frac{x_m}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x_m}{t}}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= (* z t) (- INFINITY))
    (fabs (/ (/ x_m z) t))
    (if (<= (* z t) 2e+196) (/ x_m (- y (* z t))) (/ (/ (- x_m) t) z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -((double) INFINITY)) {
		tmp = fabs(((x_m / z) / t));
	} else if ((z * t) <= 2e+196) {
		tmp = x_m / (y - (z * t));
	} else {
		tmp = (-x_m / t) / z;
	}
	return x_s * tmp;
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z * t) <= -Double.POSITIVE_INFINITY) {
		tmp = Math.abs(((x_m / z) / t));
	} else if ((z * t) <= 2e+196) {
		tmp = x_m / (y - (z * t));
	} else {
		tmp = (-x_m / t) / z;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z * t) <= -math.inf:
		tmp = math.fabs(((x_m / z) / t))
	elif (z * t) <= 2e+196:
		tmp = x_m / (y - (z * t))
	else:
		tmp = (-x_m / t) / z
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (Float64(z * t) <= Float64(-Inf))
		tmp = abs(Float64(Float64(x_m / z) / t));
	elseif (Float64(z * t) <= 2e+196)
		tmp = Float64(x_m / Float64(y - Float64(z * t)));
	else
		tmp = Float64(Float64(Float64(-x_m) / t) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z * t) <= -Inf)
		tmp = abs(((x_m / z) / t));
	elseif ((z * t) <= 2e+196)
		tmp = x_m / (y - (z * t));
	else
		tmp = (-x_m / t) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Abs[N[(N[(x$95$m / z), $MachinePrecision] / t), $MachinePrecision]], $MachinePrecision], If[LessEqual[N[(z * t), $MachinePrecision], 2e+196], N[(x$95$m / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[((-x$95$m) / t), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 56.1%

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

   3. Taylor expanded in y around 0 56.1%

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

      1. associate-*r/ 56.1%

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

      2. neg-mul-1 56.1%

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

   5. Simplified 56.1%

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

      1. associate-/r* 100.0%

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

      2. add-sqr-sqrt 73.2%

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

      3. sqrt-unprod 68.8%

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

      4. clear-num 68.7%

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

      5. clear-num 68.6%

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

      6. unpow-1 68.6%

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

      7. unpow-1 68.6%

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

      8. pow-prod-up 68.7%

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

      9. div-inv 68.7%

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

      10. clear-num 68.7%

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

      11. add-sqr-sqrt 15.0%

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

      12. sqrt-unprod 48.8%

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

      13. sqr-neg 48.8%

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

      14. sqrt-unprod 53.5%

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

      15. add-sqr-sqrt 68.7%

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

      16. metadata-eval 68.7%

          \[\leadsto \sqrt{{\left(z \cdot \frac{t}{x}\right)}^{\color{blue}{-2}}} \]

   7. Applied egg-rr 68.7%

      \[\leadsto \color{blue}{\sqrt{{\left(z \cdot \frac{t}{x}\right)}^{-2}}} \]
   8. Step-by-step derivation

      1. metadata-eval 68.7%

         \[\leadsto \sqrt{{\left(z \cdot \frac{t}{x}\right)}^{\color{blue}{\left(2 \cdot -1\right)}}} \]

      2. pow-sqr 68.6%

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

      3. rem-sqrt-square 74.4%

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

      4. unpow-1 74.4%

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

      5. *-commutative 74.4%

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

      6. associate-*l/ 56.1%

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

      7. associate-/l* 74.4%

         \[\leadsto \left|\frac{1}{\color{blue}{\frac{t}{\frac{x}{z}}}}\right| \]

      8. associate-/l* 74.7%

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

      9. *-lft-identity 74.7%

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

   9. Simplified 74.7%

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

   if -inf.0 < (*.f64 z t) < 1.9999999999999999e196

   1. Initial program 99.9%

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

   if 1.9999999999999999e196 < (*.f64 z t)

   1. Initial program 78.9%

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

   3. Taylor expanded in y around 0 78.9%

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

      1. mul-1-neg 78.9%

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

      2. associate-/r* 99.8%

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

      3. distribute-neg-frac 99.8%

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

   5. Simplified 99.8%

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

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty:\\ \;\;\;\;\left|\frac{\frac{x}{z}}{t}\right|\\ \mathbf{elif}\;z \cdot t \leq 2 \cdot 10^{+196}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 99.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+196}\right):\\ \;\;\;\;\frac{\frac{-x_m}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{y - z \cdot t}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= (* z t) (- INFINITY)) (not (<= (* z t) 2e+196)))
    (/ (/ (- x_m) t) z)
    (/ x_m (- y (* z t))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -((double) INFINITY)) || !((z * t) <= 2e+196)) {
		tmp = (-x_m / t) / z;
	} else {
		tmp = x_m / (y - (z * t));
	}
	return x_s * tmp;
}

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -Double.POSITIVE_INFINITY) || !((z * t) <= 2e+196)) {
		tmp = (-x_m / t) / z;
	} else {
		tmp = x_m / (y - (z * t));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if ((z * t) <= -math.inf) or not ((z * t) <= 2e+196):
		tmp = (-x_m / t) / z
	else:
		tmp = x_m / (y - (z * t))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= Float64(-Inf)) || !(Float64(z * t) <= 2e+196))
		tmp = Float64(Float64(Float64(-x_m) / t) / z);
	else
		tmp = Float64(x_m / Float64(y - Float64(z * t)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -Inf) || ~(((z * t) <= 2e+196)))
		tmp = (-x_m / t) / z;
	else
		tmp = x_m / (y - (z * t));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[N[(z * t), $MachinePrecision], (-Infinity)], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+196]], $MachinePrecision]], N[(N[((-x$95$m) / t), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -inf.0 or 1.9999999999999999e196 < (*.f64 z t)

   1. Initial program 70.8%

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

   3. Taylor expanded in y around 0 70.8%

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

      1. mul-1-neg 70.8%

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

      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) < 1.9999999999999999e196

   1. Initial program 99.9%

      \[\frac{x}{y - z \cdot t} \]
   2. Add Preprocessing
3. Recombined 2 regimes into one program.

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -\infty \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+196}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y - z \cdot t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 76.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23} \lor \neg \left(z \cdot t \leq 10^{+36}\right):\\ \;\;\;\;\frac{-x_m}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= (* z t) -5e+23) (not (<= (* z t) 1e+36)))
    (/ (- x_m) (* z t))
    (/ x_m y))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+23) || !((z * t) <= 1e+36)) {
		tmp = -x_m / (z * t);
	} else {
		tmp = x_m / y;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-5d+23)) .or. (.not. ((z * t) <= 1d+36))) then
        tmp = -x_m / (z * t)
    else
        tmp = x_m / y
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+23) || !((z * t) <= 1e+36)) {
		tmp = -x_m / (z * t);
	} else {
		tmp = x_m / y;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+23) or not ((z * t) <= 1e+36):
		tmp = -x_m / (z * t)
	else:
		tmp = x_m / y
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+23) || !(Float64(z * t) <= 1e+36))
		tmp = Float64(Float64(-x_m) / Float64(z * t));
	else
		tmp = Float64(x_m / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -5e+23) || ~(((z * t) <= 1e+36)))
		tmp = -x_m / (z * t);
	else
		tmp = x_m / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+23], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+36]], $MachinePrecision]], N[((-x$95$m) / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x$95$m / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.9999999999999999e23 or 1.00000000000000004e36 < (*.f64 z t)

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 0 76.8%

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

      1. associate-*r/ 76.8%

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

      2. neg-mul-1 76.8%

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

   5. Simplified 76.8%

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

   if -4.9999999999999999e23 < (*.f64 z t) < 1.00000000000000004e36

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 78.2%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23} \lor \neg \left(z \cdot t \leq 10^{+36}\right):\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 79.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23} \lor \neg \left(z \cdot t \leq 10^{+36}\right):\\ \;\;\;\;\frac{\frac{-x_m}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= (* z t) -5e+23) (not (<= (* z t) 1e+36)))
    (/ (/ (- x_m) t) z)
    (/ x_m y))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+23) || !((z * t) <= 1e+36)) {
		tmp = (-x_m / t) / z;
	} else {
		tmp = x_m / y;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-5d+23)) .or. (.not. ((z * t) <= 1d+36))) then
        tmp = (-x_m / t) / z
    else
        tmp = x_m / y
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -5e+23) || !((z * t) <= 1e+36)) {
		tmp = (-x_m / t) / z;
	} else {
		tmp = x_m / y;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if ((z * t) <= -5e+23) or not ((z * t) <= 1e+36):
		tmp = (-x_m / t) / z
	else:
		tmp = x_m / y
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -5e+23) || !(Float64(z * t) <= 1e+36))
		tmp = Float64(Float64(Float64(-x_m) / t) / z);
	else
		tmp = Float64(x_m / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -5e+23) || ~(((z * t) <= 1e+36)))
		tmp = (-x_m / t) / z;
	else
		tmp = x_m / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[N[(z * t), $MachinePrecision], -5e+23], N[Not[LessEqual[N[(z * t), $MachinePrecision], 1e+36]], $MachinePrecision]], N[(N[((-x$95$m) / t), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -4.9999999999999999e23 or 1.00000000000000004e36 < (*.f64 z t)

   1. Initial program 89.8%

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

   3. Taylor expanded in y around 0 76.8%

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

      1. mul-1-neg 76.8%

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

      2. associate-/r* 84.2%

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

      3. distribute-neg-frac 84.2%

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

   5. Simplified 84.2%

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

   if -4.9999999999999999e23 < (*.f64 z t) < 1.00000000000000004e36

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 78.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5 \cdot 10^{+23} \lor \neg \left(z \cdot t \leq 10^{+36}\right):\\ \;\;\;\;\frac{\frac{-x}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 63.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+164} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{x_m}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= (* z t) -2e+164) (not (<= (* z t) 2e+160)))
    (/ x_m (* z t))
    (/ x_m y))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+164) || !((z * t) <= 2e+160)) {
		tmp = x_m / (z * t);
	} else {
		tmp = x_m / y;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((z * t) <= (-2d+164)) .or. (.not. ((z * t) <= 2d+160))) then
        tmp = x_m / (z * t)
    else
        tmp = x_m / y
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (((z * t) <= -2e+164) || !((z * t) <= 2e+160)) {
		tmp = x_m / (z * t);
	} else {
		tmp = x_m / y;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if ((z * t) <= -2e+164) or not ((z * t) <= 2e+160):
		tmp = x_m / (z * t)
	else:
		tmp = x_m / y
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((Float64(z * t) <= -2e+164) || !(Float64(z * t) <= 2e+160))
		tmp = Float64(x_m / Float64(z * t));
	else
		tmp = Float64(x_m / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (((z * t) <= -2e+164) || ~(((z * t) <= 2e+160)))
		tmp = x_m / (z * t);
	else
		tmp = x_m / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[N[(z * t), $MachinePrecision], -2e+164], N[Not[LessEqual[N[(z * t), $MachinePrecision], 2e+160]], $MachinePrecision]], N[(x$95$m / N[(z * t), $MachinePrecision]), $MachinePrecision], N[(x$95$m / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -2e164 or 2.00000000000000001e160 < (*.f64 z t)

   1. Initial program 82.9%

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

   3. Taylor expanded in y around 0 79.2%

      \[\leadsto \color{blue}{-1 \cdot \frac{x}{t \cdot z}} \]
   4. 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%

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

   5. Simplified 79.2%

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

      1. associate-/r* 94.1%

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

      2. expm1-log1p-u 91.3%

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

      3. expm1-udef 50.8%

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

      4. associate-/r* 50.8%

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

      5. add-sqr-sqrt 17.6%

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

      6. sqrt-unprod 44.7%

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

      7. sqr-neg 44.7%

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

      8. sqrt-unprod 31.9%

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

      9. add-sqr-sqrt 48.0%

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

   7. Applied egg-rr 48.0%

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

      1. expm1-def 47.5%

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

      2. expm1-log1p 47.6%

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

      3. *-commutative 47.6%

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

   9. Simplified 47.6%

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

   if -2e164 < (*.f64 z t) < 2.00000000000000001e160

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf 65.2%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -2 \cdot 10^{+164} \lor \neg \left(z \cdot t \leq 2 \cdot 10^{+160}\right):\\ \;\;\;\;\frac{x}{z \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 55.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x_s \cdot \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{+263}:\\ \;\;\;\;\frac{x_m}{y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x_m}{t}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= t 1.2e+263) (/ x_m y) (* z (/ x_m t)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 1.2e+263) {
		tmp = x_m / y;
	} else {
		tmp = z * (x_m / t);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.2d+263) then
        tmp = x_m / y
    else
        tmp = z * (x_m / t)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= 1.2e+263) {
		tmp = x_m / y;
	} else {
		tmp = z * (x_m / t);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= 1.2e+263:
		tmp = x_m / y
	else:
		tmp = z * (x_m / t)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= 1.2e+263)
		tmp = Float64(x_m / y);
	else
		tmp = Float64(z * Float64(x_m / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= 1.2e+263)
		tmp = x_m / y;
	else
		tmp = z * (x_m / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, 1.2e+263], N[(x$95$m / y), $MachinePrecision], N[(z * N[(x$95$m / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.2e263

   1. Initial program 95.2%

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

   3. Taylor expanded in y around inf 52.6%

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

   if 1.2e263 < t

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0 76.4%

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

      1. mul-1-neg 76.4%

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

      2. associate-/r* 78.9%

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

      3. distribute-neg-frac 78.9%

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

   5. Simplified 78.9%

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

      1. add-cbrt-cube 66.8%

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

      2. pow3 66.8%

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

      3. distribute-neg-frac 66.8%

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

   7. Applied egg-rr 66.8%

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

   8. Taylor expanded in x around 0 76.4%

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

      1. *-commutative 76.4%

         \[\leadsto \frac{x \cdot \sqrt[3]{-1}}{\color{blue}{z \cdot t}} \]

      2. times-frac 81.9%

         \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sqrt[3]{-1}}{t}} \]

   10. Simplified 81.9%

       \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sqrt[3]{-1}}{t}} \]

   12. Applied egg-rr 33.1%

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

       1. associate-*l/ 48.2%

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

   14. Simplified 48.2%

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

4. Final simplification 52.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{+263}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 55.0% accurate, 2.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x_s \cdot \frac{x_m}{y} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (/ x_m y)))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / y);
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m / y)
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * (x_m / y);
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	return x_s * (x_m / y)

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * Float64(x_m / y))
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m / y);
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in y around inf 51.5%

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

4. Final simplification 51.5%

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

Alternative 9: 3.5% accurate, 7.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x_s \cdot 1 \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t) :precision binary64 (* x_s 1.0))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * 1.0;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * 1.0d0
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * 1.0;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	return x_s * 1.0

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * 1.0)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * 1.0;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * 1.0), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in y around 0 51.6%

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

   1. mul-1-neg 51.6%

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

   2. associate-/r* 55.7%

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

   3. distribute-neg-frac 55.7%

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

5. Simplified 55.7%

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

   1. add-cbrt-cube 40.7%

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

   2. pow3 40.7%

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

   3. distribute-neg-frac 40.7%

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

7. Applied egg-rr 40.7%

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

8. Taylor expanded in x around 0 51.6%

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

   1. *-commutative 51.6%

      \[\leadsto \frac{x \cdot \sqrt[3]{-1}}{\color{blue}{z \cdot t}} \]

   2. times-frac 52.8%

      \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sqrt[3]{-1}}{t}} \]

10. Simplified 52.8%

    \[\leadsto \color{blue}{\frac{x}{z} \cdot \frac{\sqrt[3]{-1}}{t}} \]

12. Applied egg-rr 0.8%

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

    1. *-inverses 3.7%

       \[\leadsto \color{blue}{1} \]

14. Simplified 3.7%

    \[\leadsto \color{blue}{1} \]

15. Final simplification 3.7%

    \[\leadsto 1 \]
16. Add Preprocessing

Alternative 10: 4.3% accurate, 7.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x_s \cdot x_m \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x_s x_m y z t) :precision binary64 (* x_s x_m))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
assert(x_m < y && y < z && z < t);
double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * x_m;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * x_m
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
assert x_m < y && y < z && z < t;
public static double code(double x_s, double x_m, double y, double z, double t) {
	return x_s * x_m;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
[x_m, y, z, t] = sort([x_m, y, z, t])
def code(x_s, x_m, y, z, t):
	return x_s * x_m

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
x_m, y, z, t = sort([x_m, y, z, t])
function code(x_s, x_m, y, z, t)
	return Float64(x_s * x_m)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
x_m, y, z, t = num2cell(sort([x_m, y, z, t])){:}
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * x_m;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
NOTE: x_m, y, z, and t should be sorted in increasing order before calling this function.
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * x$95$m), $MachinePrecision]

Derivation

1. Initial program 95.1%

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

3. Taylor expanded in y around 0 51.6%

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

   1. associate-*r/ 51.6%

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

   2. neg-mul-1 51.6%

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

5. Simplified 51.6%

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

   1. associate-/r* 55.7%

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

   2. expm1-log1p-u 47.0%

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

   3. expm1-udef 32.4%

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

   4. associate-/r* 33.1%

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

   5. add-sqr-sqrt 13.7%

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

   6. sqrt-unprod 26.5%

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

   7. sqr-neg 26.5%

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

   8. sqrt-unprod 14.9%

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

   9. add-sqr-sqrt 24.8%

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

7. Applied egg-rr 24.8%

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

   1. expm1-def 18.0%

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

   2. expm1-log1p 20.1%

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

   3. *-commutative 20.1%

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

9. Simplified 20.1%

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

11. Applied egg-rr 3.1%

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

    1. *-commutative 3.1%

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

    2. associate-*l/ 1.7%

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

    3. associate-/l* 1.3%

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

    4. *-inverses 4.2%

       \[\leadsto \frac{x}{\color{blue}{1}} \]

13. Simplified 4.2%

    \[\leadsto \color{blue}{\frac{x}{1}} \]

14. Final simplification 4.2%

    \[\leadsto x \]
15. Add Preprocessing

Developer target: 96.1% accurate, 0.3× 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 2024018 
(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))))