Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B

Percentage Accurate: 89.2% → 96.8%

Time: 8.8s

Alternatives: 11

Speedup: 0.8×

Specification

Math

\[\begin{array}{l} \\ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (* (- y z) (- t z))))

C

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

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 - z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / ((y - z) * (t - z));
}

Python

def code(x, y, z, t):
	return x / ((y - z) * (t - z))

Julia

function code(x, y, z, t)
	return Float64(x / Float64(Float64(y - z) * Float64(t - z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / ((y - z) * (t - z));
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 89.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x}{\left(y - z\right) \cdot \left(t - z\right)} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (/ x (* (- y z) (- t z))))

C

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

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 - z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x / ((y - z) * (t - z));
}

Python

def code(x, y, z, t):
	return x / ((y - z) * (t - z))

Julia

function code(x, y, z, t)
	return Float64(x / Float64(Float64(y - z) * Float64(t - z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x / ((y - z) * (t - z));
end

Wolfram

code[x_, y_, z_, t_] := N[(x / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 96.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 3.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(t - z\right)}^{-1}}{\frac{y - z}{x\_m}}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 (<= x_m 3.1e+71)
    (/ x_m (* (- t z) (- y z)))
    (/ (pow (- t z) -1.0) (/ (- y z) 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) {
	double tmp;
	if (x_m <= 3.1e+71) {
		tmp = x_m / ((t - z) * (y - z));
	} else {
		tmp = pow((t - z), -1.0) / ((y - z) / x_m);
	}
	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 (x_m <= 3.1d+71) then
        tmp = x_m / ((t - z) * (y - z))
    else
        tmp = ((t - z) ** (-1.0d0)) / ((y - z) / x_m)
    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 (x_m <= 3.1e+71) {
		tmp = x_m / ((t - z) * (y - z));
	} else {
		tmp = Math.pow((t - z), -1.0) / ((y - z) / x_m);
	}
	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 x_m <= 3.1e+71:
		tmp = x_m / ((t - z) * (y - z))
	else:
		tmp = math.pow((t - z), -1.0) / ((y - z) / x_m)
	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 (x_m <= 3.1e+71)
		tmp = Float64(x_m / Float64(Float64(t - z) * Float64(y - z)));
	else
		tmp = Float64((Float64(t - z) ^ -1.0) / Float64(Float64(y - z) / x_m));
	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 (x_m <= 3.1e+71)
		tmp = x_m / ((t - z) * (y - z));
	else
		tmp = ((t - z) ^ -1.0) / ((y - z) / x_m);
	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[x$95$m, 3.1e+71], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Power[N[(t - z), $MachinePrecision], -1.0], $MachinePrecision] / N[(N[(y - z), $MachinePrecision] / x$95$m), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 3.10000000000000018e71

   1. Initial program 97.2%

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

   if 3.10000000000000018e71 < x

   1. Initial program 77.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/r* N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. clear-num N/A

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

      12. flip-- N/A

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

      13. lift--.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 90.4

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

   4. Applied rewrites 90.4%

      \[\leadsto \color{blue}{\frac{{\left(t - z\right)}^{-1}}{\frac{y - z}{x}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 3.1 \cdot 10^{+71}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{{\left(t - z\right)}^{-1}}{\frac{y - z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 97.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;\frac{\frac{x\_m}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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
 (let* ((t_1 (/ x_m (* (- t z) (- y z)))))
   (* x_s (if (<= t_1 0.0) (/ (/ x_m (- y z)) (- t z)) t_1))))

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 t_1 = x_m / ((t - z) * (y - z));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (x_m / (y - z)) / (t - z);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x_m / ((t - z) * (y - z))
    if (t_1 <= 0.0d0) then
        tmp = (x_m / (y - z)) / (t - z)
    else
        tmp = t_1
    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 t_1 = x_m / ((t - z) * (y - z));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = (x_m / (y - z)) / (t - z);
	} else {
		tmp = t_1;
	}
	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):
	t_1 = x_m / ((t - z) * (y - z))
	tmp = 0
	if t_1 <= 0.0:
		tmp = (x_m / (y - z)) / (t - z)
	else:
		tmp = t_1
	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)
	t_1 = Float64(x_m / Float64(Float64(t - z) * Float64(y - z)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(Float64(x_m / Float64(y - z)) / Float64(t - z));
	else
		tmp = t_1;
	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)
	t_1 = x_m / ((t - z) * (y - z));
	tmp = 0.0;
	if (t_1 <= 0.0)
		tmp = (x_m / (y - z)) / (t - z);
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 0.0], N[(N[(x$95$m / N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], t$95$1]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z))) < 0.0

   1. Initial program 90.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/r* N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 96.4

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

   4. Applied rewrites 96.4%

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

   if 0.0 < (/.f64 x (*.f64 (-.f64 y z) (-.f64 t z)))

   1. Initial program 99.6%

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

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)} \leq 0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \left(t - z\right) \cdot \left(y - z\right)\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 10^{+132}:\\ \;\;\;\;\frac{x\_m}{t\_1}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x\_m}{t - z}}{y - z}\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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
 (let* ((t_1 (* (- t z) (- y z))))
   (* x_s (if (<= t_1 1e+132) (/ x_m t_1) (/ (/ x_m (- t z)) (- y 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 t_1 = (t - z) * (y - z);
	double tmp;
	if (t_1 <= 1e+132) {
		tmp = x_m / t_1;
	} else {
		tmp = (x_m / (t - z)) / (y - z);
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = (t - z) * (y - z)
    if (t_1 <= 1d+132) then
        tmp = x_m / t_1
    else
        tmp = (x_m / (t - z)) / (y - z)
    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 t_1 = (t - z) * (y - z);
	double tmp;
	if (t_1 <= 1e+132) {
		tmp = x_m / t_1;
	} else {
		tmp = (x_m / (t - z)) / (y - 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):
	t_1 = (t - z) * (y - z)
	tmp = 0
	if t_1 <= 1e+132:
		tmp = x_m / t_1
	else:
		tmp = (x_m / (t - z)) / (y - 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)
	t_1 = Float64(Float64(t - z) * Float64(y - z))
	tmp = 0.0
	if (t_1 <= 1e+132)
		tmp = Float64(x_m / t_1);
	else
		tmp = Float64(Float64(x_m / Float64(t - z)) / Float64(y - 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)
	t_1 = (t - z) * (y - z);
	tmp = 0.0;
	if (t_1 <= 1e+132)
		tmp = x_m / t_1;
	else
		tmp = (x_m / (t - z)) / (y - 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_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 1e+132], N[(x$95$m / t$95$1), $MachinePrecision], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 y z) (-.f64 t z)) < 9.99999999999999991e131

   1. Initial program 97.1%

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

   if 9.99999999999999991e131 < (*.f64 (-.f64 y z) (-.f64 t z))

   1. Initial program 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l/ N/A

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

      4. lower-/.f64 N/A

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

      5. lower-/.f64 99.6

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

   4. Applied rewrites 99.6%

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

4. Final simplification 98.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(t - z\right) \cdot \left(y - z\right) \leq 10^{+132}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 68.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -4.2:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(y - z\right)}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{x\_m}{\left(-z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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
 (let* ((t_1 (/ x_m (* z z))))
   (*
    x_s
    (if (<= z -4.2)
      t_1
      (if (<= z 2.2e-17)
        (/ x_m (* t (- y z)))
        (if (<= z 1.15e+57) (/ x_m (* (- z) y)) t_1))))))

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 t_1 = x_m / (z * z);
	double tmp;
	if (z <= -4.2) {
		tmp = t_1;
	} else if (z <= 2.2e-17) {
		tmp = x_m / (t * (y - z));
	} else if (z <= 1.15e+57) {
		tmp = x_m / (-z * y);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x_m / (z * z)
    if (z <= (-4.2d0)) then
        tmp = t_1
    else if (z <= 2.2d-17) then
        tmp = x_m / (t * (y - z))
    else if (z <= 1.15d+57) then
        tmp = x_m / (-z * y)
    else
        tmp = t_1
    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 t_1 = x_m / (z * z);
	double tmp;
	if (z <= -4.2) {
		tmp = t_1;
	} else if (z <= 2.2e-17) {
		tmp = x_m / (t * (y - z));
	} else if (z <= 1.15e+57) {
		tmp = x_m / (-z * y);
	} else {
		tmp = t_1;
	}
	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):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -4.2:
		tmp = t_1
	elif z <= 2.2e-17:
		tmp = x_m / (t * (y - z))
	elif z <= 1.15e+57:
		tmp = x_m / (-z * y)
	else:
		tmp = t_1
	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)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -4.2)
		tmp = t_1;
	elseif (z <= 2.2e-17)
		tmp = Float64(x_m / Float64(t * Float64(y - z)));
	elseif (z <= 1.15e+57)
		tmp = Float64(x_m / Float64(Float64(-z) * y));
	else
		tmp = t_1;
	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)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -4.2)
		tmp = t_1;
	elseif (z <= 2.2e-17)
		tmp = x_m / (t * (y - z));
	elseif (z <= 1.15e+57)
		tmp = x_m / (-z * y);
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -4.2], t$95$1, If[LessEqual[z, 2.2e-17], N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e+57], N[(x$95$m / N[((-z) * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -4.20000000000000018 or 1.1499999999999999e57 < z

   1. Initial program 89.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 78.7

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

   5. Applied rewrites 78.7%

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

   if -4.20000000000000018 < z < 2.2e-17

   1. Initial program 96.8%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if 2.2e-17 < z < 1.1499999999999999e57

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 47.3

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{x}{\left(-1 \cdot z\right) \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.1%

      \[\leadsto \frac{x}{\left(-z\right) \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 5: 61.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-20}:\\ \;\;\;\;\frac{x\_m}{t \cdot y}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+57}:\\ \;\;\;\;\frac{x\_m}{\left(-z\right) \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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
 (let* ((t_1 (/ x_m (* z z))))
   (*
    x_s
    (if (<= z -7.2e-70)
      t_1
      (if (<= z 7e-20)
        (/ x_m (* t y))
        (if (<= z 1.15e+57) (/ x_m (* (- z) y)) t_1))))))

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 t_1 = x_m / (z * z);
	double tmp;
	if (z <= -7.2e-70) {
		tmp = t_1;
	} else if (z <= 7e-20) {
		tmp = x_m / (t * y);
	} else if (z <= 1.15e+57) {
		tmp = x_m / (-z * y);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x_m / (z * z)
    if (z <= (-7.2d-70)) then
        tmp = t_1
    else if (z <= 7d-20) then
        tmp = x_m / (t * y)
    else if (z <= 1.15d+57) then
        tmp = x_m / (-z * y)
    else
        tmp = t_1
    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 t_1 = x_m / (z * z);
	double tmp;
	if (z <= -7.2e-70) {
		tmp = t_1;
	} else if (z <= 7e-20) {
		tmp = x_m / (t * y);
	} else if (z <= 1.15e+57) {
		tmp = x_m / (-z * y);
	} else {
		tmp = t_1;
	}
	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):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -7.2e-70:
		tmp = t_1
	elif z <= 7e-20:
		tmp = x_m / (t * y)
	elif z <= 1.15e+57:
		tmp = x_m / (-z * y)
	else:
		tmp = t_1
	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)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -7.2e-70)
		tmp = t_1;
	elseif (z <= 7e-20)
		tmp = Float64(x_m / Float64(t * y));
	elseif (z <= 1.15e+57)
		tmp = Float64(x_m / Float64(Float64(-z) * y));
	else
		tmp = t_1;
	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)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -7.2e-70)
		tmp = t_1;
	elseif (z <= 7e-20)
		tmp = x_m / (t * y);
	elseif (z <= 1.15e+57)
		tmp = x_m / (-z * y);
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -7.2e-70], t$95$1, If[LessEqual[z, 7e-20], N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e+57], N[(x$95$m / N[((-z) * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -7.2000000000000004e-70 or 1.1499999999999999e57 < z

   1. Initial program 90.5%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 73.1

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

   5. Applied rewrites 73.1%

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

   if -7.2000000000000004e-70 < z < 7.00000000000000007e-20

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 75.4

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

   5. Applied rewrites 75.4%

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

   if 7.00000000000000007e-20 < z < 1.1499999999999999e57

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 47.3

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in t around 0

      \[\leadsto \frac{x}{\left(-1 \cdot z\right) \cdot y} \]
   7. Step-by-step derivation

   8. Applied rewrites 36.1%

      \[\leadsto \frac{x}{\left(-z\right) \cdot y} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 6: 78.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-33}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot y}\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-177}:\\ \;\;\;\;\frac{x\_m}{\left(z - t\right) \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(y - z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 (<= y -1.5e-33)
    (/ x_m (* (- t z) y))
    (if (<= y 6.8e-177) (/ x_m (* (- z t) z)) (/ x_m (* t (- y 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 (y <= -1.5e-33) {
		tmp = x_m / ((t - z) * y);
	} else if (y <= 6.8e-177) {
		tmp = x_m / ((z - t) * z);
	} else {
		tmp = x_m / (t * (y - z));
	}
	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 (y <= (-1.5d-33)) then
        tmp = x_m / ((t - z) * y)
    else if (y <= 6.8d-177) then
        tmp = x_m / ((z - t) * z)
    else
        tmp = x_m / (t * (y - z))
    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 (y <= -1.5e-33) {
		tmp = x_m / ((t - z) * y);
	} else if (y <= 6.8e-177) {
		tmp = x_m / ((z - t) * z);
	} else {
		tmp = x_m / (t * (y - 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 y <= -1.5e-33:
		tmp = x_m / ((t - z) * y)
	elif y <= 6.8e-177:
		tmp = x_m / ((z - t) * z)
	else:
		tmp = x_m / (t * (y - 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 (y <= -1.5e-33)
		tmp = Float64(x_m / Float64(Float64(t - z) * y));
	elseif (y <= 6.8e-177)
		tmp = Float64(x_m / Float64(Float64(z - t) * z));
	else
		tmp = Float64(x_m / Float64(t * Float64(y - 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 (y <= -1.5e-33)
		tmp = x_m / ((t - z) * y);
	elseif (y <= 6.8e-177)
		tmp = x_m / ((z - t) * z);
	else
		tmp = x_m / (t * (y - 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[y, -1.5e-33], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.8e-177], N[(x$95$m / N[(N[(z - t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -1.5000000000000001e-33

   1. Initial program 92.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if -1.5000000000000001e-33 < y < 6.8000000000000001e-177

   1. Initial program 93.9%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. remove-double-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 78.4

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

   5. Applied rewrites 78.4%

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

   if 6.8000000000000001e-177 < y

   1. Initial program 93.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 59.9

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

   5. Applied rewrites 59.9%

      \[\leadsto \frac{x}{\color{blue}{t \cdot \left(y - z\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 70.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{-35}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot y}\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-307}:\\ \;\;\;\;\frac{x\_m}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{t \cdot \left(y - z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 (<= y -6.6e-35)
    (/ x_m (* (- t z) y))
    (if (<= y 2.4e-307) (/ x_m (* z z)) (/ x_m (* t (- y 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 (y <= -6.6e-35) {
		tmp = x_m / ((t - z) * y);
	} else if (y <= 2.4e-307) {
		tmp = x_m / (z * z);
	} else {
		tmp = x_m / (t * (y - z));
	}
	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 (y <= (-6.6d-35)) then
        tmp = x_m / ((t - z) * y)
    else if (y <= 2.4d-307) then
        tmp = x_m / (z * z)
    else
        tmp = x_m / (t * (y - z))
    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 (y <= -6.6e-35) {
		tmp = x_m / ((t - z) * y);
	} else if (y <= 2.4e-307) {
		tmp = x_m / (z * z);
	} else {
		tmp = x_m / (t * (y - 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 y <= -6.6e-35:
		tmp = x_m / ((t - z) * y)
	elif y <= 2.4e-307:
		tmp = x_m / (z * z)
	else:
		tmp = x_m / (t * (y - 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 (y <= -6.6e-35)
		tmp = Float64(x_m / Float64(Float64(t - z) * y));
	elseif (y <= 2.4e-307)
		tmp = Float64(x_m / Float64(z * z));
	else
		tmp = Float64(x_m / Float64(t * Float64(y - 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 (y <= -6.6e-35)
		tmp = x_m / ((t - z) * y);
	elseif (y <= 2.4e-307)
		tmp = x_m / (z * z);
	else
		tmp = x_m / (t * (y - 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[y, -6.6e-35], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.4e-307], N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -6.6000000000000001e-35

   1. Initial program 92.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 85.9

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

   5. Applied rewrites 85.9%

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

   if -6.6000000000000001e-35 < y < 2.40000000000000018e-307

   1. Initial program 96.4%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 64.1

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

   5. Applied rewrites 64.1%

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

   if 2.40000000000000018e-307 < y

   1. Initial program 92.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 59.0

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

   5. Applied rewrites 59.0%

      \[\leadsto \frac{x}{\color{blue}{t \cdot \left(y - z\right)}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 90.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 -6.5e+166) (/ (/ x_m z) (- z t)) (/ x_m (* (- t z) (- y 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 <= -6.5e+166) {
		tmp = (x_m / z) / (z - t);
	} else {
		tmp = x_m / ((t - z) * (y - z));
	}
	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 <= (-6.5d+166)) then
        tmp = (x_m / z) / (z - t)
    else
        tmp = x_m / ((t - z) * (y - z))
    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 <= -6.5e+166) {
		tmp = (x_m / z) / (z - t);
	} else {
		tmp = x_m / ((t - z) * (y - 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 <= -6.5e+166:
		tmp = (x_m / z) / (z - t)
	else:
		tmp = x_m / ((t - z) * (y - 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 (z <= -6.5e+166)
		tmp = Float64(Float64(x_m / z) / Float64(z - t));
	else
		tmp = Float64(x_m / Float64(Float64(t - z) * Float64(y - 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 <= -6.5e+166)
		tmp = (x_m / z) / (z - t);
	else
		tmp = x_m / ((t - z) * (y - 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[z, -6.5e+166], N[(N[(x$95$m / z), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -6.5000000000000005e166

   1. Initial program 81.5%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. associate-/r* N/A

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

      3. distribute-neg-frac2 N/A

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

      4. mul-1-neg N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. mul-1-neg N/A

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

      8. sub-neg N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. remove-double-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 99.9

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

   5. Applied rewrites 99.9%

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

   if -6.5000000000000005e166 < z

   1. Initial program 95.4%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{x}{z}}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 61.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ \begin{array}{l} t_1 := \frac{x\_m}{z \cdot z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.25 \cdot 10^{-16}:\\ \;\;\;\;\frac{x\_m}{t \cdot y}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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
 (let* ((t_1 (/ x_m (* z z))))
   (* x_s (if (<= z -7.2e-70) t_1 (if (<= z 4.25e-16) (/ x_m (* t y)) t_1)))))

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 t_1 = x_m / (z * z);
	double tmp;
	if (z <= -7.2e-70) {
		tmp = t_1;
	} else if (z <= 4.25e-16) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = x_m / (z * z)
    if (z <= (-7.2d-70)) then
        tmp = t_1
    else if (z <= 4.25d-16) then
        tmp = x_m / (t * y)
    else
        tmp = t_1
    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 t_1 = x_m / (z * z);
	double tmp;
	if (z <= -7.2e-70) {
		tmp = t_1;
	} else if (z <= 4.25e-16) {
		tmp = x_m / (t * y);
	} else {
		tmp = t_1;
	}
	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):
	t_1 = x_m / (z * z)
	tmp = 0
	if z <= -7.2e-70:
		tmp = t_1
	elif z <= 4.25e-16:
		tmp = x_m / (t * y)
	else:
		tmp = t_1
	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)
	t_1 = Float64(x_m / Float64(z * z))
	tmp = 0.0
	if (z <= -7.2e-70)
		tmp = t_1;
	elseif (z <= 4.25e-16)
		tmp = Float64(x_m / Float64(t * y));
	else
		tmp = t_1;
	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)
	t_1 = x_m / (z * z);
	tmp = 0.0;
	if (z <= -7.2e-70)
		tmp = t_1;
	elseif (z <= 4.25e-16)
		tmp = x_m / (t * y);
	else
		tmp = t_1;
	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_] := Block[{t$95$1 = N[(x$95$m / N[(z * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -7.2e-70], t$95$1, If[LessEqual[z, 4.25e-16], N[(x$95$m / N[(t * y), $MachinePrecision]), $MachinePrecision], t$95$1]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.2000000000000004e-70 or 4.25e-16 < z

   1. Initial program 90.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \frac{x}{\color{blue}{{z}^{2}}} \]
   4. Step-by-step derivation

      1. unpow2 N/A

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

      2. lower-*.f64 67.8

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

   5. Applied rewrites 67.8%

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

   if -7.2000000000000004e-70 < z < 4.25e-16

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 74.8

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

   5. Applied rewrites 74.8%

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

Alternative 10: 90.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ [x_m, y, z, t] = \mathsf{sort}([x_m, y, z, t])\\ \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{x\_m}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\left(t - z\right) \cdot \left(y - z\right)}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 -6.5e+166) (/ (/ x_m z) z) (/ x_m (* (- t z) (- y 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 <= -6.5e+166) {
		tmp = (x_m / z) / z;
	} else {
		tmp = x_m / ((t - z) * (y - z));
	}
	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 <= (-6.5d+166)) then
        tmp = (x_m / z) / z
    else
        tmp = x_m / ((t - z) * (y - z))
    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 <= -6.5e+166) {
		tmp = (x_m / z) / z;
	} else {
		tmp = x_m / ((t - z) * (y - 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 <= -6.5e+166:
		tmp = (x_m / z) / z
	else:
		tmp = x_m / ((t - z) * (y - 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 (z <= -6.5e+166)
		tmp = Float64(Float64(x_m / z) / z);
	else
		tmp = Float64(x_m / Float64(Float64(t - z) * Float64(y - 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 <= -6.5e+166)
		tmp = (x_m / z) / z;
	else
		tmp = x_m / ((t - z) * (y - 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[z, -6.5e+166], N[(N[(x$95$m / z), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m / N[(N[(t - z), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -6.5000000000000005e166

   1. Initial program 81.5%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-/l* N/A

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

      6. associate-/r* N/A

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

      7. lift--.f64 N/A

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

      8. flip-- N/A

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

      9. clear-num N/A

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

      10. lower-/.f64 N/A

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

      11. clear-num N/A

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

      12. flip-- N/A

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

      13. lift--.f64 N/A

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

      14. inv-pow N/A

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

      15. lower-pow.f64 N/A

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

      16. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

      \[\leadsto \color{blue}{\frac{x}{{z}^{2}}} \]
   6. Step-by-step derivation

      1. unpow2 N/A

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

      2. associate-/r* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 95.1

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

   7. Applied rewrites 95.1%

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

   if -6.5000000000000005e166 < z

   1. Initial program 95.4%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+166}:\\ \;\;\;\;\frac{\frac{x}{z}}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(t - z\right) \cdot \left(y - z\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 39.0% accurate, 1.4× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 (* t 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 / (t * 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 / (t * 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 / (t * 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 / (t * 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 / Float64(t * 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 / (t * 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 / N[(t * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 93.3%

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

3. Taylor expanded in z around 0

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

   1. lower-*.f64 45.1

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

5. Applied rewrites 45.1%

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

Developer Target 1: 87.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot \left(t - z\right)\\ \mathbf{if}\;\frac{x}{t\_1} < 0:\\ \;\;\;\;\frac{\frac{x}{y - z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{1}{t\_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) (- t z))))
   (if (< (/ x t_1) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if ((x / t_1) < 0.0) {
		tmp = (x / (y - z)) / (t - z);
	} else {
		tmp = x * (1.0 / 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 = (y - z) * (t - z)
    if ((x / t_1) < 0.0d0) then
        tmp = (x / (y - z)) / (t - z)
    else
        tmp = x * (1.0d0 / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * (t - z);
	double tmp;
	if ((x / t_1) < 0.0) {
		tmp = (x / (y - z)) / (t - z);
	} else {
		tmp = x * (1.0 / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y - z) * (t - z)
	tmp = 0
	if (x / t_1) < 0.0:
		tmp = (x / (y - z)) / (t - z)
	else:
		tmp = x * (1.0 / t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * Float64(t - z))
	tmp = 0.0
	if (Float64(x / t_1) < 0.0)
		tmp = Float64(Float64(x / Float64(y - z)) / Float64(t - z));
	else
		tmp = Float64(x * Float64(1.0 / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y - z) * (t - z);
	tmp = 0.0;
	if ((x / t_1) < 0.0)
		tmp = (x / (y - z)) / (t - z);
	else
		tmp = x * (1.0 / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]}, If[Less[N[(x / t$95$1), $MachinePrecision], 0.0], N[(N[(x / N[(y - z), $MachinePrecision]), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(x * N[(1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024255 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (/ x (* (- y z) (- t z))) 0) (/ (/ x (- y z)) (- t z)) (* x (/ 1 (* (- y z) (- t z))))))

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