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

Percentage Accurate: 95.6% → 97.8%

Time: 9.1s

Alternatives: 7

Speedup: 1.0×

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: 97.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+277}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{t} \cdot \frac{-1}{z}\\ \end{array} \end{array} \]

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 (<= (* z t) 5e+277) (/ x (fma (- z) t y)) (* (/ x t) (/ -1.0 z))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double tmp;
	if ((z * t) <= 5e+277) {
		tmp = x / fma(-z, t, y);
	} else {
		tmp = (x / t) * (-1.0 / z);
	}
	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) <= 5e+277)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(Float64(x / t) * Float64(-1.0 / z));
	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], 5e+277], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], N[(N[(x / t), $MachinePrecision] * N[(-1.0 / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 4.99999999999999982e277

   1. Initial program 98.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 98.5

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

   4. Applied egg-rr 98.5%

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

   if 4.99999999999999982e277 < (*.f64 z t)

   1. Initial program 64.0%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 64.0

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

   5. Simplified 64.0%

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

      1. associate-/r* N/A

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

      2. div-inv N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. metadata-eval N/A

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

      6. frac-2neg N/A

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

      7. /-lowering-/.f64 99.9

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

   7. Applied egg-rr 99.9%

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

Alternative 2: 77.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x}{z \cdot \left(-t\right)}\\ \mathbf{if}\;z \cdot t \leq -5:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

NOTE: x, y, z, and t should be sorted in increasing order before calling this function.
(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ x (* z (- t)))))
   (if (<= (* z t) -5.0) t_1 (if (<= (* z t) 5e-80) (/ x y) t_1))))

C

assert(x < y && y < z && z < t);
double code(double x, double y, double z, double t) {
	double t_1 = x / (z * -t);
	double tmp;
	if ((z * t) <= -5.0) {
		tmp = t_1;
	} else if ((z * t) <= 5e-80) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x / (z * -t)
    if ((z * t) <= (-5.0d0)) then
        tmp = t_1
    else if ((z * t) <= 5d-80) then
        tmp = x / y
    else
        tmp = t_1
    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 t_1 = x / (z * -t);
	double tmp;
	if ((z * t) <= -5.0) {
		tmp = t_1;
	} else if ((z * t) <= 5e-80) {
		tmp = x / y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

[x, y, z, t] = sort([x, y, z, t])
def code(x, y, z, t):
	t_1 = x / (z * -t)
	tmp = 0
	if (z * t) <= -5.0:
		tmp = t_1
	elif (z * t) <= 5e-80:
		tmp = x / y
	else:
		tmp = t_1
	return tmp

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	t_1 = Float64(x / Float64(z * Float64(-t)))
	tmp = 0.0
	if (Float64(z * t) <= -5.0)
		tmp = t_1;
	elseif (Float64(z * t) <= 5e-80)
		tmp = Float64(x / y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp_2 = code(x, y, z, t)
	t_1 = x / (z * -t);
	tmp = 0.0;
	if ((z * t) <= -5.0)
		tmp = t_1;
	elseif ((z * t) <= 5e-80)
		tmp = x / y;
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x / N[(z * (-t)), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(z * t), $MachinePrecision], -5.0], t$95$1, If[LessEqual[N[(z * t), $MachinePrecision], 5e-80], N[(x / y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < -5 or 5e-80 < (*.f64 z t)

   1. Initial program 92.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 73.8

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

   5. Simplified 73.8%

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

   if -5 < (*.f64 z t) < 5e-80

   1. Initial program 99.9%

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

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{\frac{x}{y}} \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 89.9

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

   5. Simplified 89.9%

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

4. Final simplification 80.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq -5:\\ \;\;\;\;\frac{x}{z \cdot \left(-t\right)}\\ \mathbf{elif}\;z \cdot t \leq 5 \cdot 10^{-80}:\\ \;\;\;\;\frac{x}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(-t\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 97.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \end{array} \end{array} \]

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) 5e+179) (/ 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) <= 5e+179) {
		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) <= 5e+179)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(-Float64(Float64(x / 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], 5e+179], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], (-N[(N[(x / z), $MachinePrecision] / t), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 5e179

   1. Initial program 98.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 98.4

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

   4. Applied egg-rr 98.4%

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

   if 5e179 < (*.f64 z t)

   1. Initial program 78.2%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 78.2

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

   5. Simplified 78.2%

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

      1. *-commutative N/A

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

      2. associate-/r* N/A

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

      3. /-lowering-/.f64 N/A

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

      4. frac-2neg N/A

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

      5. remove-double-neg N/A

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

      6. /-lowering-/.f64 N/A

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

      7. neg-lowering-neg.f64 97.5

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

   7. Applied egg-rr 97.5%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 5 \cdot 10^{+179}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{z}}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 97.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} [x, y, z, t] = \mathsf{sort}([x, y, z, t])\\ \\ \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+253}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \end{array} \]

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 (<= (* z t) 1e+253) (/ x (fma (- z) t 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 ((z * t) <= 1e+253) {
		tmp = x / fma(-z, t, 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 (Float64(z * t) <= 1e+253)
		tmp = Float64(x / fma(Float64(-z), t, y));
	else
		tmp = Float64(-Float64(Float64(x / t) / z));
	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], 1e+253], N[(x / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision], (-N[(N[(x / t), $MachinePrecision] / z), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if (*.f64 z t) < 9.9999999999999994e252

   1. Initial program 98.4%

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

      1. sub-neg N/A

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

      2. +-commutative N/A

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

      3. distribute-lft-neg-in N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. neg-lowering-neg.f64 98.4

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

   4. Applied egg-rr 98.4%

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

   if 9.9999999999999994e252 < (*.f64 z t)

   1. Initial program 69.8%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. /-lowering-/.f64 N/A

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

      5. mul-1-neg N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. mul-1-neg N/A

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

      8. *-lowering-*.f64 N/A

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

      9. mul-1-neg N/A

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

      10. neg-lowering-neg.f64 69.8

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

   5. Simplified 69.8%

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

      1. associate-/r* N/A

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

      2. frac-2neg N/A

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

      3. distribute-frac-neg N/A

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

      4. frac-2neg N/A

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

      5. /-lowering-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. remove-double-neg N/A

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

      8. /-lowering-/.f64 N/A

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

      9. neg-lowering-neg.f64 99.8

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

   7. Applied egg-rr 99.8%

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

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \cdot t \leq 10^{+253}:\\ \;\;\;\;\frac{x}{\mathsf{fma}\left(-z, t, y\right)}\\ \mathbf{else}:\\ \;\;\;\;-\frac{\frac{x}{t}}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 95.7% accurate, 1.0× speedup

Math

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

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 (fma (- z) t y)))

C

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

Julia

x, y, z, t = sort([x, y, z, t])
function code(x, y, z, t)
	return Float64(x / fma(Float64(-z), t, 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 / N[((-z) * t + y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

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

   1. sub-neg N/A

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

   2. +-commutative N/A

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

   3. distribute-lft-neg-in N/A

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

   4. accelerator-lowering-fma.f64 N/A

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

   5. neg-lowering-neg.f64 95.6

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

4. Applied egg-rr 95.6%

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

Alternative 6: 95.6% accurate, 1.0× speedup

Math

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

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 (* z t))))

C

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

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 - (z * t))
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 - (z * t));
}

Python

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

Julia

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

MATLAB

x, y, z, t = num2cell(sort([x, y, z, t])){:}
function tmp = code(x, y, z, t)
	tmp = x / (y - (z * t));
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 / N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.6%

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

Alternative 7: 54.7% accurate, 1.7× 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 95.6%

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

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{\frac{x}{y}} \]
4. Step-by-step derivation

   1. /-lowering-/.f64 54.1

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

5. Simplified 54.1%

   \[\leadsto \color{blue}{\frac{x}{y}} \]
6. Add Preprocessing

Developer Target 1: 96.7% accurate, 0.4× 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 2024198 
(FPCore (x y z t)
  :name "Diagrams.Solve.Tridiagonal:solveTriDiagonal from diagrams-solve-0.1, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -161819597360704900000000000000000000000000000000000) (/ 1 (- (/ y x) (* (/ z x) t))) (if (< x 213783064348764440000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (/ x (- y (* z t))) (/ 1 (- (/ y x) (* (/ z x) t))))))

  (/ x (- y (* z t))))