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

Percentage Accurate: 89.1% → 98.3%

Time: 13.3s

Alternatives: 19

Speedup: 1.0×

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: 89.1% 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: 98.3% accurate, 0.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (* (/ (sqrt x_m) (- y z)) (/ (sqrt x_m) (- t z)))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 * ((sqrt(x_m) / (y - z)) * (sqrt(x_m) / (t - z)))
end function

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * ((sqrt(x_m) / (y - z)) * (sqrt(x_m) / (t - z)));
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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(N[(N[Sqrt[x$95$m], $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision] * N[(N[Sqrt[x$95$m], $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.8%

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

   1. add-sqr-sqrt 43.1%

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

   2. times-frac 47.8%

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

3. Applied egg-rr 47.8%

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

4. Final simplification 47.8%

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

Alternative 2: 48.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{\frac{x_m}{t}}{y}\\ t_2 := \frac{-x_m}{y \cdot z}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+190}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+130}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq -4 \cdot 10^{+98}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{x_m}{y \cdot t}\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-140} \lor \neg \left(y \leq 7 \cdot 10^{-27}\right):\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{-x_m}{z \cdot t}\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ (/ x_m t) y)) (t_2 (/ (- x_m) (* y z))))
   (*
    x_s
    (if (<= y -1.22e+190)
      t_2
      (if (<= y -2.7e+130)
        t_1
        (if (<= y -4e+98)
          t_2
          (if (<= y -8.2e-111)
            (/ x_m (* y t))
            (if (or (<= y -1.2e-140) (not (<= y 7e-27)))
              t_1
              (/ (- x_m) (* z t))))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / t) / y;
	double t_2 = -x_m / (y * z);
	double tmp;
	if (y <= -1.22e+190) {
		tmp = t_2;
	} else if (y <= -2.7e+130) {
		tmp = t_1;
	} else if (y <= -4e+98) {
		tmp = t_2;
	} else if (y <= -8.2e-111) {
		tmp = x_m / (y * t);
	} else if ((y <= -1.2e-140) || !(y <= 7e-27)) {
		tmp = t_1;
	} else {
		tmp = -x_m / (z * t);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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) :: t_2
    real(8) :: tmp
    t_1 = (x_m / t) / y
    t_2 = -x_m / (y * z)
    if (y <= (-1.22d+190)) then
        tmp = t_2
    else if (y <= (-2.7d+130)) then
        tmp = t_1
    else if (y <= (-4d+98)) then
        tmp = t_2
    else if (y <= (-8.2d-111)) then
        tmp = x_m / (y * t)
    else if ((y <= (-1.2d-140)) .or. (.not. (y <= 7d-27))) then
        tmp = t_1
    else
        tmp = -x_m / (z * t)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (x_m / t) / y;
	double t_2 = -x_m / (y * z);
	double tmp;
	if (y <= -1.22e+190) {
		tmp = t_2;
	} else if (y <= -2.7e+130) {
		tmp = t_1;
	} else if (y <= -4e+98) {
		tmp = t_2;
	} else if (y <= -8.2e-111) {
		tmp = x_m / (y * t);
	} else if ((y <= -1.2e-140) || !(y <= 7e-27)) {
		tmp = t_1;
	} else {
		tmp = -x_m / (z * t);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (x_m / t) / y
	t_2 = -x_m / (y * z)
	tmp = 0
	if y <= -1.22e+190:
		tmp = t_2
	elif y <= -2.7e+130:
		tmp = t_1
	elif y <= -4e+98:
		tmp = t_2
	elif y <= -8.2e-111:
		tmp = x_m / (y * t)
	elif (y <= -1.2e-140) or not (y <= 7e-27):
		tmp = t_1
	else:
		tmp = -x_m / (z * t)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(x_m / t) / y)
	t_2 = Float64(Float64(-x_m) / Float64(y * z))
	tmp = 0.0
	if (y <= -1.22e+190)
		tmp = t_2;
	elseif (y <= -2.7e+130)
		tmp = t_1;
	elseif (y <= -4e+98)
		tmp = t_2;
	elseif (y <= -8.2e-111)
		tmp = Float64(x_m / Float64(y * t));
	elseif ((y <= -1.2e-140) || !(y <= 7e-27))
		tmp = t_1;
	else
		tmp = Float64(Float64(-x_m) / Float64(z * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (x_m / t) / y;
	t_2 = -x_m / (y * z);
	tmp = 0.0;
	if (y <= -1.22e+190)
		tmp = t_2;
	elseif (y <= -2.7e+130)
		tmp = t_1;
	elseif (y <= -4e+98)
		tmp = t_2;
	elseif (y <= -8.2e-111)
		tmp = x_m / (y * t);
	elseif ((y <= -1.2e-140) || ~((y <= 7e-27)))
		tmp = t_1;
	else
		tmp = -x_m / (z * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision]}, Block[{t$95$2 = N[((-x$95$m) / N[(y * z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[y, -1.22e+190], t$95$2, If[LessEqual[y, -2.7e+130], t$95$1, If[LessEqual[y, -4e+98], t$95$2, If[LessEqual[y, -8.2e-111], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -1.2e-140], N[Not[LessEqual[y, 7e-27]], $MachinePrecision]], t$95$1, N[((-x$95$m) / N[(z * t), $MachinePrecision]), $MachinePrecision]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.21999999999999995e190 or -2.6999999999999998e130 < y < -3.99999999999999999e98

   1. Initial program 99.5%

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

   2. Taylor expanded in y around inf 92.5%

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

      1. *-commutative 92.5%

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

      2. associate-/r* 93.1%

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

   4. Simplified 93.1%

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

   5. Taylor expanded in t around 0 69.1%

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

      1. associate-*r/ 69.1%

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

      2. neg-mul-1 69.1%

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

      3. *-commutative 69.1%

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

   7. Simplified 69.1%

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

   if -1.21999999999999995e190 < y < -2.6999999999999998e130 or -8.19999999999999936e-111 < y < -1.19999999999999993e-140 or 7.0000000000000003e-27 < y

   1. Initial program 84.4%

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

   2. Taylor expanded in y around inf 71.4%

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

      1. *-commutative 71.4%

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

      2. associate-/r* 78.3%

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

   4. Simplified 78.3%

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

   5. Taylor expanded in t around inf 49.6%

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

   if -3.99999999999999999e98 < y < -8.19999999999999936e-111

   1. Initial program 95.6%

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

   2. Taylor expanded in z around 0 37.8%

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

   if -1.19999999999999993e-140 < y < 7.0000000000000003e-27

   1. Initial program 91.8%

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

      1. *-un-lft-identity 91.8%

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

      2. times-frac 95.6%

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

   3. Applied egg-rr 95.6%

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

   4. Taylor expanded in y around 0 72.5%

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

      1. mul-1-neg 72.5%

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

      2. associate-/r* 77.4%

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

      3. distribute-neg-frac 77.4%

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

      4. distribute-neg-frac 77.4%

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

   6. Simplified 77.4%

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

   7. Taylor expanded in z around 0 43.5%

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

      1. associate-*r/ 43.5%

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

      2. neg-mul-1 43.5%

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

   9. Simplified 43.5%

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

4. Final simplification 47.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.22 \cdot 10^{+190}:\\ \;\;\;\;\frac{-x}{y \cdot z}\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{+130}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{elif}\;y \leq -4 \cdot 10^{+98}:\\ \;\;\;\;\frac{-x}{y \cdot z}\\ \mathbf{elif}\;y \leq -8.2 \cdot 10^{-111}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{-140} \lor \neg \left(y \leq 7 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 3: 66.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{x_m}{z \cdot \left(t - z\right)}\\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+80}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\frac{x_m}{\left(y - z\right) \cdot t}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+131}:\\ \;\;\;\;-\frac{\frac{x_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ x_m (* z (- t z)))))
   (*
    x_s
    (if (<= z -8.8e+80)
      t_1
      (if (<= z 3e-9)
        (/ x_m (* (- y z) t))
        (if (<= z 3.4e+131) (- (/ (/ x_m z) y)) t_1))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * (t - z));
	double tmp;
	if (z <= -8.8e+80) {
		tmp = t_1;
	} else if (z <= 3e-9) {
		tmp = x_m / ((y - z) * t);
	} else if (z <= 3.4e+131) {
		tmp = -((x_m / z) / y);
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 * (t - z))
    if (z <= (-8.8d+80)) then
        tmp = t_1
    else if (z <= 3d-9) then
        tmp = x_m / ((y - z) * t)
    else if (z <= 3.4d+131) 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);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m / (z * (t - z));
	double tmp;
	if (z <= -8.8e+80) {
		tmp = t_1;
	} else if (z <= 3e-9) {
		tmp = x_m / ((y - z) * t);
	} else if (z <= 3.4e+131) {
		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)
def code(x_s, x_m, y, z, t):
	t_1 = x_m / (z * (t - z))
	tmp = 0
	if z <= -8.8e+80:
		tmp = t_1
	elif z <= 3e-9:
		tmp = x_m / ((y - z) * t)
	elif z <= 3.4e+131:
		tmp = -((x_m / z) / y)
	else:
		tmp = t_1
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m / Float64(z * Float64(t - z)))
	tmp = 0.0
	if (z <= -8.8e+80)
		tmp = t_1;
	elseif (z <= 3e-9)
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	elseif (z <= 3.4e+131)
		tmp = Float64(-Float64(Float64(x_m / 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);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m / (z * (t - z));
	tmp = 0.0;
	if (z <= -8.8e+80)
		tmp = t_1;
	elseif (z <= 3e-9)
		tmp = x_m / ((y - z) * t);
	elseif (z <= 3.4e+131)
		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]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m / N[(z * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -8.8e+80], t$95$1, If[LessEqual[z, 3e-9], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.4e+131], (-N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -8.80000000000000011e80 or 3.39999999999999986e131 < z

   1. Initial program 81.6%

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

      1. *-un-lft-identity 81.6%

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

      2. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 80.3%

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

      1. mul-1-neg 80.3%

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

      2. associate-/r* 92.2%

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

      3. distribute-neg-frac 92.2%

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

      4. distribute-neg-frac 92.2%

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

   6. Simplified 92.2%

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

      1. expm1-log1p-u 91.0%

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

      2. expm1-udef 71.4%

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

      3. associate-/l/ 71.4%

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

      4. add-sqr-sqrt 42.2%

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

      5. sqrt-unprod 67.2%

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

      6. sqr-neg 67.2%

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

      7. sqrt-unprod 28.2%

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

      8. add-sqr-sqrt 70.4%

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

      9. *-commutative 70.4%

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

   8. Applied egg-rr 70.4%

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

      1. expm1-def 70.1%

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

      2. expm1-log1p 70.1%

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

   10. Simplified 70.1%

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

   if -8.80000000000000011e80 < z < 2.99999999999999998e-9

   1. Initial program 96.9%

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

   2. Taylor expanded in t around inf 71.8%

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

   if 2.99999999999999998e-9 < z < 3.39999999999999986e131

   1. Initial program 88.9%

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

   2. Taylor expanded in y around inf 32.4%

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

      1. *-commutative 32.4%

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

      2. associate-/r* 45.9%

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

   4. Simplified 45.9%

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

   5. Taylor expanded in t around 0 29.5%

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

      1. associate-*r/ 29.5%

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

      2. neg-mul-1 29.5%

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

   7. Simplified 29.5%

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

4. Final simplification 65.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+80}:\\ \;\;\;\;\frac{x}{z \cdot \left(t - z\right)}\\ \mathbf{elif}\;z \leq 3 \cdot 10^{-9}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{+131}:\\ \;\;\;\;-\frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z \cdot \left(t - z\right)}\\ \end{array} \]

Alternative 4: 93.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+112} \lor \neg \left(z \leq 5.5 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{\frac{-x_m}{z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -1.32e+112) (not (<= z 5.5e+136)))
    (/ (/ (- x_m) z) (- t z))
    (/ x_m (* (- y z) (- t z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -1.32e+112) || !(z <= 5.5e+136)) {
		tmp = (-x_m / z) / (t - z);
	} else {
		tmp = x_m / ((y - z) * (t - z));
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 <= (-1.32d+112)) .or. (.not. (z <= 5.5d+136))) then
        tmp = (-x_m / z) / (t - z)
    else
        tmp = x_m / ((y - z) * (t - z))
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -1.32e+112) || !(z <= 5.5e+136)) {
		tmp = (-x_m / z) / (t - z);
	} else {
		tmp = x_m / ((y - z) * (t - z));
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -1.32e+112) or not (z <= 5.5e+136):
		tmp = (-x_m / z) / (t - z)
	else:
		tmp = x_m / ((y - z) * (t - z))
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -1.32e+112) || !(z <= 5.5e+136))
		tmp = Float64(Float64(Float64(-x_m) / z) / Float64(t - z));
	else
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -1.32e+112) || ~((z <= 5.5e+136)))
		tmp = (-x_m / z) / (t - z);
	else
		tmp = x_m / ((y - z) * (t - z));
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -1.32e+112], N[Not[LessEqual[z, 5.5e+136]], $MachinePrecision]], N[(N[((-x$95$m) / z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.32e112 or 5.50000000000000039e136 < z

   1. Initial program 80.2%

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

      1. *-un-lft-identity 80.2%

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

      2. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 80.0%

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

      1. mul-1-neg 80.0%

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

      2. associate-/r* 92.9%

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

      3. distribute-neg-frac 92.9%

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

      4. distribute-neg-frac 92.9%

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

   6. Simplified 92.9%

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

   if -1.32e112 < z < 5.50000000000000039e136

   1. Initial program 95.4%

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

4. Final simplification 94.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.32 \cdot 10^{+112} \lor \neg \left(z \leq 5.5 \cdot 10^{+136}\right):\\ \;\;\;\;\frac{\frac{-x}{z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \end{array} \]

Alternative 5: 93.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{-x_m}{z}}{t - z}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{x_m}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{y + z}}{z + t}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -2.45e+110)
    (/ (/ (- x_m) z) (- t z))
    (if (<= z 5.6e+123)
      (/ x_m (* (- y z) (- t z)))
      (/ (/ x_m (+ y z)) (+ z t))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -2.45e+110) {
		tmp = (-x_m / z) / (t - z);
	} else if (z <= 5.6e+123) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (x_m / (y + z)) / (z + t);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 <= (-2.45d+110)) then
        tmp = (-x_m / z) / (t - z)
    else if (z <= 5.6d+123) then
        tmp = x_m / ((y - z) * (t - z))
    else
        tmp = (x_m / (y + z)) / (z + t)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -2.45e+110) {
		tmp = (-x_m / z) / (t - z);
	} else if (z <= 5.6e+123) {
		tmp = x_m / ((y - z) * (t - z));
	} else {
		tmp = (x_m / (y + z)) / (z + t);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -2.45e+110:
		tmp = (-x_m / z) / (t - z)
	elif z <= 5.6e+123:
		tmp = x_m / ((y - z) * (t - z))
	else:
		tmp = (x_m / (y + z)) / (z + t)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -2.45e+110)
		tmp = Float64(Float64(Float64(-x_m) / z) / Float64(t - z));
	elseif (z <= 5.6e+123)
		tmp = Float64(x_m / Float64(Float64(y - z) * Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / Float64(y + z)) / Float64(z + t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -2.45e+110)
		tmp = (-x_m / z) / (t - z);
	elseif (z <= 5.6e+123)
		tmp = x_m / ((y - z) * (t - z));
	else
		tmp = (x_m / (y + z)) / (z + t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -2.45e+110], N[(N[((-x$95$m) / z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.6e+123], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / N[(y + z), $MachinePrecision]), $MachinePrecision] / N[(z + t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -2.45000000000000001e110

   1. Initial program 74.3%

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

      1. *-un-lft-identity 74.3%

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

      2. times-frac 99.9%

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

   3. Applied egg-rr 99.9%

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

   4. Taylor expanded in y around 0 74.1%

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

      1. mul-1-neg 74.1%

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

      2. associate-/r* 89.7%

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

      3. distribute-neg-frac 89.7%

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

      4. distribute-neg-frac 89.7%

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

   6. Simplified 89.7%

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

   if -2.45000000000000001e110 < z < 5.60000000000000023e123

   1. Initial program 96.4%

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

   if 5.60000000000000023e123 < z

   1. Initial program 82.1%

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

      1. add-sqr-sqrt 29.5%

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

      2. times-frac 37.6%

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

   3. Applied egg-rr 37.6%

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

      1. associate-*r/ 37.7%

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

      2. associate-*l/ 37.7%

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

      3. add-sqr-sqrt 99.9%

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

      4. sub-neg 99.9%

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

      5. add-sqr-sqrt 0.0%

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

      6. sqrt-unprod 77.5%

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

      7. sqr-neg 77.5%

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

      8. sqrt-unprod 77.2%

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

      9. add-sqr-sqrt 77.2%

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

      10. sub-neg 77.2%

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

      11. add-sqr-sqrt 0.0%

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

      12. sqrt-unprod 84.0%

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

      13. sqr-neg 84.0%

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

      14. sqrt-unprod 95.0%

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

      15. add-sqr-sqrt 95.1%

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

   5. Applied egg-rr 95.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.45 \cdot 10^{+110}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t - z}\\ \mathbf{elif}\;z \leq 5.6 \cdot 10^{+123}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{y + z}}{z + t}\\ \end{array} \]

Alternative 6: 71.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{x_m}{t - z}}{y}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{-x_m}{z \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -7.5e-16)
    (/ (/ x_m (- t z)) y)
    (if (<= y 6.5e-88) (/ (- x_m) (* z (- t z))) (/ (/ x_m t) (- y z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -7.5e-16) {
		tmp = (x_m / (t - z)) / y;
	} else if (y <= 6.5e-88) {
		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)
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 <= (-7.5d-16)) then
        tmp = (x_m / (t - z)) / y
    else if (y <= 6.5d-88) 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);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -7.5e-16) {
		tmp = (x_m / (t - z)) / y;
	} else if (y <= 6.5e-88) {
		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)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -7.5e-16:
		tmp = (x_m / (t - z)) / y
	elif y <= 6.5e-88:
		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)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -7.5e-16)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	elseif (y <= 6.5e-88)
		tmp = Float64(Float64(-x_m) / Float64(z * Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -7.5e-16)
		tmp = (x_m / (t - z)) / y;
	elseif (y <= 6.5e-88)
		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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -7.5e-16], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 6.5e-88], N[((-x$95$m) / N[(z * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -7.5e-16

   1. Initial program 94.9%

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

   2. Taylor expanded in y around inf 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-/r* 85.9%

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

   4. Simplified 85.9%

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

   if -7.5e-16 < y < 6.50000000000000006e-88

   1. Initial program 92.9%

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

   2. Taylor expanded in y around 0 75.8%

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

      1. associate-*r/ 75.8%

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

      2. neg-mul-1 75.8%

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

   4. Simplified 75.8%

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

   if 6.50000000000000006e-88 < y

   1. Initial program 85.4%

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

      1. *-un-lft-identity 85.4%

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

      2. times-frac 94.5%

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

   3. Applied egg-rr 94.5%

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

      1. associate-*l/ 94.6%

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

      2. *-un-lft-identity 94.6%

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

   5. Applied egg-rr 94.6%

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

   6. Taylor expanded in t around inf 47.0%

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

      1. associate-/r* 51.0%

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

   8. Simplified 51.0%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-16}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{-88}:\\ \;\;\;\;\frac{-x}{z \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 7: 72.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{x_m}{t - z}}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{-x_m}{z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -2.25e-15)
    (/ (/ x_m (- t z)) y)
    (if (<= y 2.2e-165) (/ (/ (- x_m) z) (- t z)) (/ (/ x_m t) (- y z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -2.25e-15) {
		tmp = (x_m / (t - z)) / y;
	} else if (y <= 2.2e-165) {
		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)
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 <= (-2.25d-15)) then
        tmp = (x_m / (t - z)) / y
    else if (y <= 2.2d-165) 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);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -2.25e-15) {
		tmp = (x_m / (t - z)) / y;
	} else if (y <= 2.2e-165) {
		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)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -2.25e-15:
		tmp = (x_m / (t - z)) / y
	elif y <= 2.2e-165:
		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)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -2.25e-15)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	elseif (y <= 2.2e-165)
		tmp = Float64(Float64(Float64(-x_m) / z) / Float64(t - z));
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -2.25e-15)
		tmp = (x_m / (t - z)) / y;
	elseif (y <= 2.2e-165)
		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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -2.25e-15], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[y, 2.2e-165], N[(N[((-x$95$m) / z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -2.2499999999999999e-15

   1. Initial program 94.9%

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

   2. Taylor expanded in y around inf 85.7%

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

      1. *-commutative 85.7%

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

      2. associate-/r* 85.9%

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

   4. Simplified 85.9%

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

   if -2.2499999999999999e-15 < y < 2.1999999999999999e-165

   1. Initial program 92.8%

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

      1. *-un-lft-identity 92.8%

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

      2. times-frac 95.5%

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

   3. Applied egg-rr 95.5%

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

   4. Taylor expanded in y around 0 76.0%

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

      1. mul-1-neg 76.0%

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

      2. associate-/r* 79.8%

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

      3. distribute-neg-frac 79.8%

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

      4. distribute-neg-frac 79.8%

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

   6. Simplified 79.8%

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

   if 2.1999999999999999e-165 < y

   1. Initial program 86.6%

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

      1. *-un-lft-identity 86.6%

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

      2. times-frac 95.3%

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

   3. Applied egg-rr 95.3%

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

      1. associate-*l/ 95.3%

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

      2. *-un-lft-identity 95.3%

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

   5. Applied egg-rr 95.3%

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

   6. Taylor expanded in t around inf 47.9%

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

      1. associate-/r* 52.2%

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

   8. Simplified 52.2%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.25 \cdot 10^{-15}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-165}:\\ \;\;\;\;\frac{\frac{-x}{z}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 8: 58.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x_m}{t}}{y}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-145}:\\ \;\;\;\;\frac{-1}{y} \cdot \frac{x_m}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -6.2e-37)
    (/ (/ x_m t) y)
    (if (<= t 1.25e-145) (* (/ -1.0 y) (/ x_m z)) (/ x_m (* (- y z) t))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -6.2e-37) {
		tmp = (x_m / t) / y;
	} else if (t <= 1.25e-145) {
		tmp = (-1.0 / y) * (x_m / z);
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-6.2d-37)) then
        tmp = (x_m / t) / y
    else if (t <= 1.25d-145) then
        tmp = ((-1.0d0) / y) * (x_m / z)
    else
        tmp = x_m / ((y - z) * t)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -6.2e-37) {
		tmp = (x_m / t) / y;
	} else if (t <= 1.25e-145) {
		tmp = (-1.0 / y) * (x_m / z);
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -6.2e-37:
		tmp = (x_m / t) / y
	elif t <= 1.25e-145:
		tmp = (-1.0 / y) * (x_m / z)
	else:
		tmp = x_m / ((y - z) * t)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -6.2e-37)
		tmp = Float64(Float64(x_m / t) / y);
	elseif (t <= 1.25e-145)
		tmp = Float64(Float64(-1.0 / y) * Float64(x_m / z));
	else
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -6.2e-37)
		tmp = (x_m / t) / y;
	elseif (t <= 1.25e-145)
		tmp = (-1.0 / y) * (x_m / z);
	else
		tmp = x_m / ((y - z) * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -6.2e-37], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[t, 1.25e-145], N[(N[(-1.0 / y), $MachinePrecision] * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -6.19999999999999987e-37

   1. Initial program 88.4%

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

   2. Taylor expanded in y around inf 56.3%

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

      1. *-commutative 56.3%

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

      2. associate-/r* 58.0%

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

   4. Simplified 58.0%

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

   5. Taylor expanded in t around inf 56.2%

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

   if -6.19999999999999987e-37 < t < 1.2499999999999999e-145

   1. Initial program 94.1%

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

   2. Taylor expanded in y around inf 58.1%

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

      1. *-commutative 58.1%

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

      2. associate-/r* 61.9%

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

   4. Simplified 61.9%

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

      1. expm1-log1p-u 50.6%

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

      2. expm1-udef 36.6%

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

      3. associate-/l/ 38.4%

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

      4. sub-neg 38.4%

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

      5. add-sqr-sqrt 15.0%

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

      6. sqrt-unprod 39.7%

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

      7. sqr-neg 39.7%

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

      8. sqrt-unprod 21.0%

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

      9. add-sqr-sqrt 34.7%

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

   6. Applied egg-rr 34.7%

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

      1. expm1-def 33.5%

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

      2. expm1-log1p 38.2%

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

      3. associate-/r* 37.4%

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

   8. Simplified 37.4%

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

   9. Taylor expanded in t around 0 23.3%

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

       1. *-commutative 23.3%

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

   11. Simplified 23.3%

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

       1. *-commutative 23.3%

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

       2. add-sqr-sqrt 9.2%

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

       3. sqrt-unprod 31.8%

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

       4. sqr-neg 31.8%

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

       5. sqrt-unprod 23.6%

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

       6. add-sqr-sqrt 43.5%

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

       7. neg-mul-1 43.5%

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

       8. times-frac 47.4%

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

   13. Applied egg-rr 47.4%

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

   if 1.2499999999999999e-145 < t

   1. Initial program 88.8%

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

   2. Taylor expanded in t around inf 67.6%

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

4. Final simplification 56.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.2 \cdot 10^{-37}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{elif}\;t \leq 1.25 \cdot 10^{-145}:\\ \;\;\;\;\frac{-1}{y} \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 9: 48.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-140} \lor \neg \left(y \leq 2.6 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{\frac{x_m}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x_m}{z \cdot t}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -1.25e-140) (not (<= y 2.6e-27)))
    (/ (/ x_m t) y)
    (/ (- x_m) (* z t)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -1.25e-140) || !(y <= 2.6e-27)) {
		tmp = (x_m / t) / y;
	} else {
		tmp = -x_m / (z * t);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 <= (-1.25d-140)) .or. (.not. (y <= 2.6d-27))) then
        tmp = (x_m / t) / y
    else
        tmp = -x_m / (z * t)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -1.25e-140) || !(y <= 2.6e-27)) {
		tmp = (x_m / t) / y;
	} else {
		tmp = -x_m / (z * t);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -1.25e-140) or not (y <= 2.6e-27):
		tmp = (x_m / t) / y
	else:
		tmp = -x_m / (z * t)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -1.25e-140) || !(y <= 2.6e-27))
		tmp = Float64(Float64(x_m / t) / y);
	else
		tmp = Float64(Float64(-x_m) / Float64(z * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -1.25e-140) || ~((y <= 2.6e-27)))
		tmp = (x_m / t) / y;
	else
		tmp = -x_m / (z * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -1.25e-140], N[Not[LessEqual[y, 2.6e-27]], $MachinePrecision]], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision], N[((-x$95$m) / N[(z * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.25000000000000004e-140 or 2.60000000000000017e-27 < y

   1. Initial program 90.2%

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

   2. Taylor expanded in y around inf 69.8%

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

      1. *-commutative 69.8%

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

      2. associate-/r* 73.5%

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

   4. Simplified 73.5%

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

   5. Taylor expanded in t around inf 47.9%

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

   if -1.25000000000000004e-140 < y < 2.60000000000000017e-27

   1. Initial program 91.8%

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

      1. *-un-lft-identity 91.8%

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

      2. times-frac 95.6%

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

   3. Applied egg-rr 95.6%

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

   4. Taylor expanded in y around 0 72.2%

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

      1. mul-1-neg 72.2%

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

      2. associate-/r* 77.1%

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

      3. distribute-neg-frac 77.1%

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

      4. distribute-neg-frac 77.1%

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

   6. Simplified 77.1%

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

   7. Taylor expanded in z around 0 43.9%

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

      1. associate-*r/ 43.9%

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

      2. neg-mul-1 43.9%

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

   9. Simplified 43.9%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-140} \lor \neg \left(y \leq 2.6 \cdot 10^{-27}\right):\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{-x}{z \cdot t}\\ \end{array} \]

Alternative 10: 52.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+80} \lor \neg \left(z \leq 1.3 \cdot 10^{-11}\right):\\ \;\;\;\;-\frac{\frac{x_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{t}}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -8e+80) (not (<= z 1.3e-11)))
    (- (/ (/ x_m z) y))
    (/ (/ x_m t) y))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -8e+80) || !(z <= 1.3e-11)) {
		tmp = -((x_m / z) / y);
	} else {
		tmp = (x_m / t) / y;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 <= (-8d+80)) .or. (.not. (z <= 1.3d-11))) then
        tmp = -((x_m / z) / y)
    else
        tmp = (x_m / t) / y
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -8e+80) || !(z <= 1.3e-11)) {
		tmp = -((x_m / z) / y);
	} else {
		tmp = (x_m / t) / y;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -8e+80) or not (z <= 1.3e-11):
		tmp = -((x_m / z) / y)
	else:
		tmp = (x_m / t) / y
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -8e+80) || !(z <= 1.3e-11))
		tmp = Float64(-Float64(Float64(x_m / z) / y));
	else
		tmp = Float64(Float64(x_m / t) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -8e+80) || ~((z <= 1.3e-11)))
		tmp = -((x_m / z) / y);
	else
		tmp = (x_m / t) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -8e+80], N[Not[LessEqual[z, 1.3e-11]], $MachinePrecision]], (-N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision]), N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -8e80 or 1.3e-11 < z

   1. Initial program 83.7%

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

   2. Taylor expanded in y around inf 38.2%

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

      1. *-commutative 38.2%

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

      2. associate-/r* 52.3%

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

   4. Simplified 52.3%

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

   5. Taylor expanded in t around 0 46.6%

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

      1. associate-*r/ 46.6%

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

      2. neg-mul-1 46.6%

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

   7. Simplified 46.6%

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

   if -8e80 < z < 1.3e-11

   1. Initial program 96.9%

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

   2. Taylor expanded in y around inf 71.2%

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

      1. *-commutative 71.2%

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

      2. associate-/r* 69.5%

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

   4. Simplified 69.5%

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

   5. Taylor expanded in t around inf 56.3%

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

4. Final simplification 51.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8 \cdot 10^{+80} \lor \neg \left(z \leq 1.3 \cdot 10^{-11}\right):\\ \;\;\;\;-\frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 11: 46.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+80} \lor \neg \left(z \leq 0.04\right):\\ \;\;\;\;\frac{x_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{y \cdot t}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -9e+80) (not (<= z 0.04))) (/ x_m (* y z)) (/ x_m (* y t)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -9e+80) || !(z <= 0.04)) {
		tmp = x_m / (y * z);
	} else {
		tmp = x_m / (y * t);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 <= (-9d+80)) .or. (.not. (z <= 0.04d0))) then
        tmp = x_m / (y * z)
    else
        tmp = x_m / (y * t)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -9e+80) || !(z <= 0.04)) {
		tmp = x_m / (y * z);
	} else {
		tmp = x_m / (y * t);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -9e+80) or not (z <= 0.04):
		tmp = x_m / (y * z)
	else:
		tmp = x_m / (y * t)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -9e+80) || !(z <= 0.04))
		tmp = Float64(x_m / Float64(y * z));
	else
		tmp = Float64(x_m / Float64(y * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -9e+80) || ~((z <= 0.04)))
		tmp = x_m / (y * z);
	else
		tmp = x_m / (y * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -9e+80], N[Not[LessEqual[z, 0.04]], $MachinePrecision]], N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -9.00000000000000013e80 or 0.0400000000000000008 < z

   1. Initial program 83.3%

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

   2. Taylor expanded in y around inf 38.3%

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

      1. *-commutative 38.3%

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

      2. associate-/r* 52.7%

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

   4. Simplified 52.7%

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

      1. expm1-log1p-u 52.5%

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

      2. expm1-udef 54.9%

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

      3. associate-/l/ 54.9%

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

      4. sub-neg 54.9%

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

      5. add-sqr-sqrt 20.8%

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

      6. sqrt-unprod 60.2%

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

      7. sqr-neg 60.2%

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

      8. sqrt-unprod 33.9%

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

      9. add-sqr-sqrt 54.7%

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

   6. Applied egg-rr 54.7%

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

      1. expm1-def 36.2%

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

      2. expm1-log1p 36.5%

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

      3. associate-/r* 38.4%

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

   8. Simplified 38.4%

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

   9. Taylor expanded in t around 0 34.8%

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

       1. *-commutative 34.8%

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

   11. Simplified 34.8%

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

   if -9.00000000000000013e80 < z < 0.0400000000000000008

   1. Initial program 97.0%

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

   2. Taylor expanded in z around 0 54.9%

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

4. Final simplification 45.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{+80} \lor \neg \left(z \leq 0.04\right):\\ \;\;\;\;\frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y \cdot t}\\ \end{array} \]

Alternative 12: 50.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+80} \lor \neg \left(z \leq 1.76 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{\frac{x_m}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{t}}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -9.5e+80) (not (<= z 1.76e+105)))
    (/ (/ x_m z) y)
    (/ (/ x_m t) y))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -9.5e+80) || !(z <= 1.76e+105)) {
		tmp = (x_m / z) / y;
	} else {
		tmp = (x_m / t) / y;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 <= (-9.5d+80)) .or. (.not. (z <= 1.76d+105))) then
        tmp = (x_m / z) / y
    else
        tmp = (x_m / t) / y
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -9.5e+80) || !(z <= 1.76e+105)) {
		tmp = (x_m / z) / y;
	} else {
		tmp = (x_m / t) / y;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -9.5e+80) or not (z <= 1.76e+105):
		tmp = (x_m / z) / y
	else:
		tmp = (x_m / t) / y
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -9.5e+80) || !(z <= 1.76e+105))
		tmp = Float64(Float64(x_m / z) / y);
	else
		tmp = Float64(Float64(x_m / t) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -9.5e+80) || ~((z <= 1.76e+105)))
		tmp = (x_m / z) / y;
	else
		tmp = (x_m / t) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -9.5e+80], N[Not[LessEqual[z, 1.76e+105]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] / y), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -9.499999999999999e80 or 1.76e105 < z

   1. Initial program 81.4%

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

   2. Taylor expanded in y around inf 40.7%

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

      1. *-commutative 40.7%

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

      2. associate-/r* 55.5%

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

   4. Simplified 55.5%

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

      1. expm1-log1p-u 54.3%

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

      2. expm1-udef 68.2%

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

      3. sub-neg 68.2%

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

      4. add-sqr-sqrt 29.0%

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

      5. sqrt-unprod 68.1%

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

      6. sqr-neg 68.1%

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

      7. sqrt-unprod 38.9%

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

      8. add-sqr-sqrt 68.1%

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

   6. Applied egg-rr 68.1%

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

      1. expm1-def 50.4%

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

      2. expm1-log1p 50.7%

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

   8. Simplified 50.7%

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

   9. Taylor expanded in t around 0 47.5%

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

   if -9.499999999999999e80 < z < 1.76e105

   1. Initial program 96.1%

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

   2. Taylor expanded in y around inf 64.5%

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

      1. *-commutative 64.5%

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

      2. associate-/r* 64.9%

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

   4. Simplified 64.9%

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

   5. Taylor expanded in t around inf 51.6%

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

4. Final simplification 50.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+80} \lor \neg \left(z \leq 1.76 \cdot 10^{+105}\right):\\ \;\;\;\;\frac{\frac{x}{z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 13: 62.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x_m}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x_m}{\left(y - z\right) \cdot t}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= y -1.5e-109) (/ x_m (* y (- t z))) (/ x_m (* (- y z) t)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.5e-109) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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-109)) then
        tmp = x_m / (y * (t - z))
    else
        tmp = x_m / ((y - z) * t)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -1.5e-109) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = x_m / ((y - z) * t);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -1.5e-109:
		tmp = x_m / (y * (t - z))
	else:
		tmp = x_m / ((y - z) * t)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -1.5e-109)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	else
		tmp = Float64(x_m / Float64(Float64(y - z) * t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -1.5e-109)
		tmp = x_m / (y * (t - z));
	else
		tmp = x_m / ((y - z) * t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -1.5e-109], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m / N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.50000000000000011e-109

   1. Initial program 96.3%

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

   2. Taylor expanded in y around inf 71.3%

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

      1. *-commutative 71.3%

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

   4. Simplified 71.3%

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

   if -1.50000000000000011e-109 < y

   1. Initial program 88.1%

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

   2. Taylor expanded in t around inf 53.1%

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

4. Final simplification 59.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-109}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\left(y - z\right) \cdot t}\\ \end{array} \]

Alternative 14: 63.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{x_m}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= y -6.5e-112) (/ x_m (* y (- t z))) (/ (/ x_m t) (- y z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -6.5e-112) {
		tmp = x_m / (y * (t - z));
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 <= (-6.5d-112)) then
        tmp = x_m / (y * (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);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -6.5e-112) {
		tmp = x_m / (y * (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)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -6.5e-112:
		tmp = x_m / (y * (t - z))
	else:
		tmp = (x_m / t) / (y - z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -6.5e-112)
		tmp = Float64(x_m / Float64(y * Float64(t - z)));
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -6.5e-112)
		tmp = x_m / (y * (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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -6.5e-112], N[(x$95$m / N[(y * N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -6.49999999999999956e-112

   1. Initial program 96.3%

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

   2. Taylor expanded in y around inf 70.6%

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

      1. *-commutative 70.6%

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

   4. Simplified 70.6%

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

   if -6.49999999999999956e-112 < y

   1. Initial program 88.0%

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

      1. *-un-lft-identity 88.0%

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

      2. times-frac 95.3%

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

   3. Applied egg-rr 95.3%

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

      1. associate-*l/ 95.4%

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

      2. *-un-lft-identity 95.4%

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

   5. Applied egg-rr 95.4%

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

   6. Taylor expanded in t around inf 53.4%

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

      1. associate-/r* 56.6%

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

   8. Simplified 56.6%

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

4. Final simplification 61.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{x}{y \cdot \left(t - z\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 15: 64.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{x_m}{y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= y -5.5e-112) (/ (/ x_m y) (- t z)) (/ (/ x_m t) (- y z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -5.5e-112) {
		tmp = (x_m / y) / (t - z);
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 <= (-5.5d-112)) then
        tmp = (x_m / y) / (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);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -5.5e-112) {
		tmp = (x_m / y) / (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)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -5.5e-112:
		tmp = (x_m / y) / (t - z)
	else:
		tmp = (x_m / t) / (y - z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -5.5e-112)
		tmp = Float64(Float64(x_m / y) / Float64(t - z));
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -5.5e-112)
		tmp = (x_m / y) / (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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -5.5e-112], N[(N[(x$95$m / y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -5.5e-112

   1. Initial program 96.3%

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

      1. add-sqr-sqrt 53.3%

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

      2. times-frac 54.4%

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

   3. Applied egg-rr 54.4%

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

   4. Taylor expanded in y around inf 70.6%

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

      1. associate-/r* 69.0%

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

   6. Simplified 69.0%

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

   if -5.5e-112 < y

   1. Initial program 88.0%

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

      1. *-un-lft-identity 88.0%

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

      2. times-frac 95.3%

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

   3. Applied egg-rr 95.3%

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

      1. associate-*l/ 95.4%

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

      2. *-un-lft-identity 95.4%

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

   5. Applied egg-rr 95.4%

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

   6. Taylor expanded in t around inf 53.4%

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

      1. associate-/r* 56.6%

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

   8. Simplified 56.6%

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

4. Final simplification 60.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.5 \cdot 10^{-112}:\\ \;\;\;\;\frac{\frac{x}{y}}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 16: 63.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{x_m}{t - z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{t}}{y - z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= y -7.8e-171) (/ (/ x_m (- t z)) y) (/ (/ x_m t) (- y z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -7.8e-171) {
		tmp = (x_m / (t - z)) / y;
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 <= (-7.8d-171)) then
        tmp = (x_m / (t - z)) / y
    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);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -7.8e-171) {
		tmp = (x_m / (t - z)) / y;
	} else {
		tmp = (x_m / t) / (y - z);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -7.8e-171:
		tmp = (x_m / (t - z)) / y
	else:
		tmp = (x_m / t) / (y - z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -7.8e-171)
		tmp = Float64(Float64(x_m / Float64(t - z)) / y);
	else
		tmp = Float64(Float64(x_m / t) / Float64(y - z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -7.8e-171)
		tmp = (x_m / (t - z)) / y;
	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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -7.8e-171], N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -7.7999999999999997e-171

   1. Initial program 96.8%

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

   2. Taylor expanded in y around inf 64.8%

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

      1. *-commutative 64.8%

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

      2. associate-/r* 70.4%

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

   4. Simplified 70.4%

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

   if -7.7999999999999997e-171 < y

   1. Initial program 87.0%

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

      1. *-un-lft-identity 87.0%

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

      2. times-frac 95.0%

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

   3. Applied egg-rr 95.0%

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

      1. associate-*l/ 95.1%

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

      2. *-un-lft-identity 95.1%

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

   5. Applied egg-rr 95.1%

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

   6. Taylor expanded in t around inf 51.4%

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

      1. associate-/r* 54.9%

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

   8. Simplified 54.9%

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

4. Final simplification 60.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.8 \cdot 10^{-171}:\\ \;\;\;\;\frac{\frac{x}{t - z}}{y}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y - z}\\ \end{array} \]

Alternative 17: 96.7% accurate, 1.0× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (/ (/ x_m (- t z)) (- y z))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 - z)) / (y - z))
end function

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * ((x_m / (t - z)) / (y - z));
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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(N[(x$95$m / N[(t - z), $MachinePrecision]), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.8%

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

   1. *-un-lft-identity 90.8%

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

   2. times-frac 96.3%

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

3. Applied egg-rr 96.3%

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

   1. associate-*l/ 96.4%

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

   2. *-un-lft-identity 96.4%

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

5. Applied egg-rr 96.4%

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

6. Final simplification 96.4%

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

Alternative 18: 45.7% accurate, 1.3× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+81}:\\ \;\;\;\;\frac{x_m}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x_m}{t}}{y}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= z -1.05e+81) (/ x_m (* y z)) (/ (/ x_m t) y))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.05e+81) {
		tmp = x_m / (y * z);
	} else {
		tmp = (x_m / t) / y;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
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 <= (-1.05d+81)) then
        tmp = x_m / (y * z)
    else
        tmp = (x_m / t) / y
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.05e+81) {
		tmp = x_m / (y * z);
	} else {
		tmp = (x_m / t) / y;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -1.05e+81:
		tmp = x_m / (y * z)
	else:
		tmp = (x_m / t) / y
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1.05e+81)
		tmp = Float64(x_m / Float64(y * z));
	else
		tmp = Float64(Float64(x_m / t) / y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -1.05e+81)
		tmp = x_m / (y * z);
	else
		tmp = (x_m / t) / y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1.05e+81], N[(x$95$m / N[(y * z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m / t), $MachinePrecision] / y), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -1.0499999999999999e81

   1. Initial program 78.0%

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

   2. Taylor expanded in y around inf 35.7%

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

      1. *-commutative 35.7%

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

      2. associate-/r* 49.1%

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

   4. Simplified 49.1%

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

      1. expm1-log1p-u 49.1%

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

      2. expm1-udef 57.5%

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

      3. associate-/l/ 57.5%

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

      4. sub-neg 57.5%

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

      5. add-sqr-sqrt 57.5%

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

      6. sqrt-unprod 64.3%

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

      7. sqr-neg 64.3%

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

      8. sqrt-unprod 0.0%

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

      9. add-sqr-sqrt 57.4%

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

   6. Applied egg-rr 57.4%

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

      1. expm1-def 33.7%

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

      2. expm1-log1p 34.0%

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

      3. associate-/r* 32.3%

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

   8. Simplified 32.3%

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

   9. Taylor expanded in t around 0 33.8%

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

       1. *-commutative 33.8%

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

   11. Simplified 33.8%

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

   if -1.0499999999999999e81 < z

   1. Initial program 93.3%

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

   2. Taylor expanded in y around inf 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-/r* 63.9%

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

   4. Simplified 63.9%

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

   5. Taylor expanded in t around inf 45.9%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+81}:\\ \;\;\;\;\frac{x}{y \cdot z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{x}{t}}{y}\\ \end{array} \]

Alternative 19: 39.5% accurate, 1.8× speedup

Math

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

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t) :precision binary64 (* x_s (/ x_m (* y t))))

C

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

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x_s * (x_m / (y * t))
end function

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m / (y * t));
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]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * N[(x$95$m / N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 90.8%

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

2. Taylor expanded in z around 0 38.1%

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

3. Final simplification 38.1%

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

Developer target: 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 2023321 
(FPCore (x y z t)
  :name "Data.Random.Distribution.Triangular:triangularCDF from random-fu-0.2.6.2, B"
  :precision binary64

  :herbie-target
  (if (< (/ x (* (- y z) (- t z))) 0.0) (/ (/ x (- y z)) (- t z)) (* x (/ 1.0 (* (- y z) (- t z)))))

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