Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1

Percentage Accurate: 97.2% → 96.7%

Time: 11.8s

Alternatives: 17

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x - y) / (z - y)) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = ((x - y) / (z - y)) * t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 4.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{z - y} \cdot \frac{t\_m}{\frac{1}{x - y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{\frac{z - y}{x - y}}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 4.1e+37)
    (* (/ 1.0 (- z y)) (/ t_m (/ 1.0 (- x y))))
    (/ t_m (/ (- z y) (- x y))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 4.1e+37) {
		tmp = (1.0 / (z - y)) * (t_m / (1.0 / (x - y)));
	} else {
		tmp = t_m / ((z - y) / (x - y));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 4.1d+37) then
        tmp = (1.0d0 / (z - y)) * (t_m / (1.0d0 / (x - y)))
    else
        tmp = t_m / ((z - y) / (x - y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 4.1e+37) {
		tmp = (1.0 / (z - y)) * (t_m / (1.0 / (x - y)));
	} else {
		tmp = t_m / ((z - y) / (x - y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if t_m <= 4.1e+37:
		tmp = (1.0 / (z - y)) * (t_m / (1.0 / (x - y)))
	else:
		tmp = t_m / ((z - y) / (x - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 4.1e+37)
		tmp = Float64(Float64(1.0 / Float64(z - y)) * Float64(t_m / Float64(1.0 / Float64(x - y))));
	else
		tmp = Float64(t_m / Float64(Float64(z - y) / Float64(x - y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 4.1e+37)
		tmp = (1.0 / (z - y)) * (t_m / (1.0 / (x - y)));
	else
		tmp = t_m / ((z - y) / (x - y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 4.1e+37], N[(N[(1.0 / N[(z - y), $MachinePrecision]), $MachinePrecision] * N[(t$95$m / N[(1.0 / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$m / N[(N[(z - y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 4.0999999999999998e37

   1. Initial program 94.6%

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

      1. associate-*l/ 89.5%

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

      2. associate-/l* 81.7%

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

   3. Simplified 81.7%

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

      1. associate-*r/ 89.5%

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

      2. associate-*l/ 94.6%

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

      3. *-commutative 94.6%

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

      4. clear-num 93.9%

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

      5. un-div-inv 95.0%

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

   6. Applied egg-rr 95.0%

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

      1. *-un-lft-identity 95.0%

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

      2. div-inv 94.9%

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

      3. times-frac 89.5%

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

   8. Applied egg-rr 89.5%

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

   if 4.0999999999999998e37 < t

   1. Initial program 98.3%

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

      1. associate-*l/ 68.1%

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

      2. associate-/l* 93.7%

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

   3. Simplified 93.7%

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

      1. associate-*r/ 68.1%

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

      2. associate-*l/ 98.3%

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

      3. *-commutative 98.3%

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

      4. clear-num 98.3%

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

      5. un-div-inv 98.3%

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

   6. Applied egg-rr 98.3%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 4.1 \cdot 10^{+37}:\\ \;\;\;\;\frac{1}{z - y} \cdot \frac{t}{\frac{1}{x - y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x - y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \left(1 - \frac{x}{y}\right)\\ t_3 := t\_m \cdot \frac{x}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{+66}:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+33}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-79}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-278}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+45}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (- 1.0 (/ x y)))) (t_3 (* t_m (/ x (- z y)))))
   (*
    t_s
    (if (<= y -7e+111)
      t_2
      (if (<= y -1.75e+66)
        (* t_m (/ (- x y) z))
        (if (<= y -5.2e+33)
          t_2
          (if (<= y -4.2e-79)
            t_3
            (if (<= y 1.15e-278)
              (* (- x y) (/ t_m z))
              (if (<= y 4.4e+45) t_3 t_2)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (1.0 - (x / y));
	double t_3 = t_m * (x / (z - y));
	double tmp;
	if (y <= -7e+111) {
		tmp = t_2;
	} else if (y <= -1.75e+66) {
		tmp = t_m * ((x - y) / z);
	} else if (y <= -5.2e+33) {
		tmp = t_2;
	} else if (y <= -4.2e-79) {
		tmp = t_3;
	} else if (y <= 1.15e-278) {
		tmp = (x - y) * (t_m / z);
	} else if (y <= 4.4e+45) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = t_m * (1.0d0 - (x / y))
    t_3 = t_m * (x / (z - y))
    if (y <= (-7d+111)) then
        tmp = t_2
    else if (y <= (-1.75d+66)) then
        tmp = t_m * ((x - y) / z)
    else if (y <= (-5.2d+33)) then
        tmp = t_2
    else if (y <= (-4.2d-79)) then
        tmp = t_3
    else if (y <= 1.15d-278) then
        tmp = (x - y) * (t_m / z)
    else if (y <= 4.4d+45) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (1.0 - (x / y));
	double t_3 = t_m * (x / (z - y));
	double tmp;
	if (y <= -7e+111) {
		tmp = t_2;
	} else if (y <= -1.75e+66) {
		tmp = t_m * ((x - y) / z);
	} else if (y <= -5.2e+33) {
		tmp = t_2;
	} else if (y <= -4.2e-79) {
		tmp = t_3;
	} else if (y <= 1.15e-278) {
		tmp = (x - y) * (t_m / z);
	} else if (y <= 4.4e+45) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = t_m * (1.0 - (x / y))
	t_3 = t_m * (x / (z - y))
	tmp = 0
	if y <= -7e+111:
		tmp = t_2
	elif y <= -1.75e+66:
		tmp = t_m * ((x - y) / z)
	elif y <= -5.2e+33:
		tmp = t_2
	elif y <= -4.2e-79:
		tmp = t_3
	elif y <= 1.15e-278:
		tmp = (x - y) * (t_m / z)
	elif y <= 4.4e+45:
		tmp = t_3
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m * Float64(1.0 - Float64(x / y)))
	t_3 = Float64(t_m * Float64(x / Float64(z - y)))
	tmp = 0.0
	if (y <= -7e+111)
		tmp = t_2;
	elseif (y <= -1.75e+66)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (y <= -5.2e+33)
		tmp = t_2;
	elseif (y <= -4.2e-79)
		tmp = t_3;
	elseif (y <= 1.15e-278)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (y <= 4.4e+45)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = t_m * (1.0 - (x / y));
	t_3 = t_m * (x / (z - y));
	tmp = 0.0;
	if (y <= -7e+111)
		tmp = t_2;
	elseif (y <= -1.75e+66)
		tmp = t_m * ((x - y) / z);
	elseif (y <= -5.2e+33)
		tmp = t_2;
	elseif (y <= -4.2e-79)
		tmp = t_3;
	elseif (y <= 1.15e-278)
		tmp = (x - y) * (t_m / z);
	elseif (y <= 4.4e+45)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[y, -7e+111], t$95$2, If[LessEqual[y, -1.75e+66], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -5.2e+33], t$95$2, If[LessEqual[y, -4.2e-79], t$95$3, If[LessEqual[y, 1.15e-278], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.4e+45], t$95$3, t$95$2]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.0000000000000004e111 or -1.7499999999999999e66 < y < -5.1999999999999995e33 or 4.4000000000000001e45 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 83.5%

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

      1. mul-1-neg 83.5%

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

      2. div-sub 83.5%

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

      3. sub-neg 83.5%

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

      4. *-inverses 83.5%

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

      5. metadata-eval 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in x around 0 76.8%

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

      1. *-rgt-identity 76.8%

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

      2. *-commutative 76.8%

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

      3. associate-/l* 83.5%

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

      4. associate-*r* 83.5%

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

      5. *-commutative 83.5%

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

      6. distribute-lft-in 83.5%

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

      7. mul-1-neg 83.5%

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

      8. sub-neg 83.5%

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

   8. Simplified 83.5%

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

   if -7.0000000000000004e111 < y < -1.7499999999999999e66

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 80.8%

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

   if -5.1999999999999995e33 < y < -4.1999999999999999e-79 or 1.15000000000000001e-278 < y < 4.4000000000000001e45

   1. Initial program 95.0%

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

   3. Taylor expanded in x around inf 76.1%

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

   if -4.1999999999999999e-79 < y < 1.15000000000000001e-278

   1. Initial program 85.8%

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

      1. associate-*l/ 92.7%

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

      2. associate-/l* 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in z around inf 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-/l* 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+111}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -1.75 \cdot 10^{+66}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq -5.2 \cdot 10^{+33}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -4.2 \cdot 10^{-79}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;y \leq 1.15 \cdot 10^{-278}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;y \leq 4.4 \cdot 10^{+45}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \left(1 - \frac{x}{y}\right)\\ t_3 := \frac{t\_m}{\frac{z - y}{x}}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+66}:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{+32}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -4.95 \cdot 10^{-138}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-277}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+38}:\\ \;\;\;\;t\_3\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (- 1.0 (/ x y)))) (t_3 (/ t_m (/ (- z y) x))))
   (*
    t_s
    (if (<= y -7.4e+111)
      t_2
      (if (<= y -3.3e+66)
        (* t_m (/ (- x y) z))
        (if (<= y -1.6e+32)
          t_2
          (if (<= y -4.95e-138)
            t_3
            (if (<= y 2.05e-277)
              (* (- x y) (/ t_m z))
              (if (<= y 2.2e+38) t_3 t_2)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (1.0 - (x / y));
	double t_3 = t_m / ((z - y) / x);
	double tmp;
	if (y <= -7.4e+111) {
		tmp = t_2;
	} else if (y <= -3.3e+66) {
		tmp = t_m * ((x - y) / z);
	} else if (y <= -1.6e+32) {
		tmp = t_2;
	} else if (y <= -4.95e-138) {
		tmp = t_3;
	} else if (y <= 2.05e-277) {
		tmp = (x - y) * (t_m / z);
	} else if (y <= 2.2e+38) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = t_m * (1.0d0 - (x / y))
    t_3 = t_m / ((z - y) / x)
    if (y <= (-7.4d+111)) then
        tmp = t_2
    else if (y <= (-3.3d+66)) then
        tmp = t_m * ((x - y) / z)
    else if (y <= (-1.6d+32)) then
        tmp = t_2
    else if (y <= (-4.95d-138)) then
        tmp = t_3
    else if (y <= 2.05d-277) then
        tmp = (x - y) * (t_m / z)
    else if (y <= 2.2d+38) then
        tmp = t_3
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (1.0 - (x / y));
	double t_3 = t_m / ((z - y) / x);
	double tmp;
	if (y <= -7.4e+111) {
		tmp = t_2;
	} else if (y <= -3.3e+66) {
		tmp = t_m * ((x - y) / z);
	} else if (y <= -1.6e+32) {
		tmp = t_2;
	} else if (y <= -4.95e-138) {
		tmp = t_3;
	} else if (y <= 2.05e-277) {
		tmp = (x - y) * (t_m / z);
	} else if (y <= 2.2e+38) {
		tmp = t_3;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = t_m * (1.0 - (x / y))
	t_3 = t_m / ((z - y) / x)
	tmp = 0
	if y <= -7.4e+111:
		tmp = t_2
	elif y <= -3.3e+66:
		tmp = t_m * ((x - y) / z)
	elif y <= -1.6e+32:
		tmp = t_2
	elif y <= -4.95e-138:
		tmp = t_3
	elif y <= 2.05e-277:
		tmp = (x - y) * (t_m / z)
	elif y <= 2.2e+38:
		tmp = t_3
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m * Float64(1.0 - Float64(x / y)))
	t_3 = Float64(t_m / Float64(Float64(z - y) / x))
	tmp = 0.0
	if (y <= -7.4e+111)
		tmp = t_2;
	elseif (y <= -3.3e+66)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (y <= -1.6e+32)
		tmp = t_2;
	elseif (y <= -4.95e-138)
		tmp = t_3;
	elseif (y <= 2.05e-277)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (y <= 2.2e+38)
		tmp = t_3;
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = t_m * (1.0 - (x / y));
	t_3 = t_m / ((z - y) / x);
	tmp = 0.0;
	if (y <= -7.4e+111)
		tmp = t_2;
	elseif (y <= -3.3e+66)
		tmp = t_m * ((x - y) / z);
	elseif (y <= -1.6e+32)
		tmp = t_2;
	elseif (y <= -4.95e-138)
		tmp = t_3;
	elseif (y <= 2.05e-277)
		tmp = (x - y) * (t_m / z);
	elseif (y <= 2.2e+38)
		tmp = t_3;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m / N[(N[(z - y), $MachinePrecision] / x), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[y, -7.4e+111], t$95$2, If[LessEqual[y, -3.3e+66], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.6e+32], t$95$2, If[LessEqual[y, -4.95e-138], t$95$3, If[LessEqual[y, 2.05e-277], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e+38], t$95$3, t$95$2]]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.4000000000000005e111 or -3.3000000000000001e66 < y < -1.5999999999999999e32 or 2.20000000000000006e38 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 83.5%

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

      1. mul-1-neg 83.5%

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

      2. div-sub 83.5%

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

      3. sub-neg 83.5%

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

      4. *-inverses 83.5%

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

      5. metadata-eval 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in x around 0 76.8%

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

      1. *-rgt-identity 76.8%

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

      2. *-commutative 76.8%

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

      3. associate-/l* 83.5%

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

      4. associate-*r* 83.5%

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

      5. *-commutative 83.5%

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

      6. distribute-lft-in 83.5%

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

      7. mul-1-neg 83.5%

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

      8. sub-neg 83.5%

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

   8. Simplified 83.5%

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

   if -7.4000000000000005e111 < y < -3.3000000000000001e66

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 80.8%

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

   if -1.5999999999999999e32 < y < -4.95e-138 or 2.04999999999999994e-277 < y < 2.20000000000000006e38

   1. Initial program 94.4%

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

      1. associate-*l/ 95.6%

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

      2. associate-/l* 90.3%

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

   3. Simplified 90.3%

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

      1. associate-*r/ 95.6%

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

      2. associate-*l/ 94.4%

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

      3. *-commutative 94.4%

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

      4. clear-num 94.4%

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

      5. un-div-inv 96.0%

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

   6. Applied egg-rr 96.0%

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

   7. Taylor expanded in x around inf 75.7%

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

   if -4.95e-138 < y < 2.04999999999999994e-277

   1. Initial program 85.1%

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

      1. associate-*l/ 93.2%

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

      2. associate-/l* 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in z around inf 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/l* 86.5%

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

   7. Simplified 86.5%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+111}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -3.3 \cdot 10^{+66}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq -1.6 \cdot 10^{+32}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -4.95 \cdot 10^{-138}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x}}\\ \mathbf{elif}\;y \leq 2.05 \cdot 10^{-277}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{+38}:\\ \;\;\;\;\frac{t}{\frac{z - y}{x}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 87.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \left(x - y\right) \cdot \frac{t\_m}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+94}:\\ \;\;\;\;t\_m \cdot \frac{y}{y - z}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-265}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-151}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+254}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m}{1 - \frac{z}{y}}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* (- x y) (/ t_m (- z y)))))
   (*
    t_s
    (if (<= y -5.6e+94)
      (* t_m (/ y (- y z)))
      (if (<= y 3.7e-265)
        t_2
        (if (<= y 1.25e-151)
          (/ (* t_m x) (- z y))
          (if (<= y 5.2e+254) t_2 (/ t_m (- 1.0 (/ z y))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) * (t_m / (z - y));
	double tmp;
	if (y <= -5.6e+94) {
		tmp = t_m * (y / (y - z));
	} else if (y <= 3.7e-265) {
		tmp = t_2;
	} else if (y <= 1.25e-151) {
		tmp = (t_m * x) / (z - y);
	} else if (y <= 5.2e+254) {
		tmp = t_2;
	} else {
		tmp = t_m / (1.0 - (z / y));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) * (t_m / (z - y))
    if (y <= (-5.6d+94)) then
        tmp = t_m * (y / (y - z))
    else if (y <= 3.7d-265) then
        tmp = t_2
    else if (y <= 1.25d-151) then
        tmp = (t_m * x) / (z - y)
    else if (y <= 5.2d+254) then
        tmp = t_2
    else
        tmp = t_m / (1.0d0 - (z / y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) * (t_m / (z - y));
	double tmp;
	if (y <= -5.6e+94) {
		tmp = t_m * (y / (y - z));
	} else if (y <= 3.7e-265) {
		tmp = t_2;
	} else if (y <= 1.25e-151) {
		tmp = (t_m * x) / (z - y);
	} else if (y <= 5.2e+254) {
		tmp = t_2;
	} else {
		tmp = t_m / (1.0 - (z / y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) * (t_m / (z - y))
	tmp = 0
	if y <= -5.6e+94:
		tmp = t_m * (y / (y - z))
	elif y <= 3.7e-265:
		tmp = t_2
	elif y <= 1.25e-151:
		tmp = (t_m * x) / (z - y)
	elif y <= 5.2e+254:
		tmp = t_2
	else:
		tmp = t_m / (1.0 - (z / y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)))
	tmp = 0.0
	if (y <= -5.6e+94)
		tmp = Float64(t_m * Float64(y / Float64(y - z)));
	elseif (y <= 3.7e-265)
		tmp = t_2;
	elseif (y <= 1.25e-151)
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	elseif (y <= 5.2e+254)
		tmp = t_2;
	else
		tmp = Float64(t_m / Float64(1.0 - Float64(z / y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) * (t_m / (z - y));
	tmp = 0.0;
	if (y <= -5.6e+94)
		tmp = t_m * (y / (y - z));
	elseif (y <= 3.7e-265)
		tmp = t_2;
	elseif (y <= 1.25e-151)
		tmp = (t_m * x) / (z - y);
	elseif (y <= 5.2e+254)
		tmp = t_2;
	else
		tmp = t_m / (1.0 - (z / y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[y, -5.6e+94], N[(t$95$m * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 3.7e-265], t$95$2, If[LessEqual[y, 1.25e-151], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 5.2e+254], t$95$2, N[(t$95$m / N[(1.0 - N[(z / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -5.59999999999999997e94

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 93.3%

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

      1. neg-mul-1 93.3%

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

      2. distribute-neg-frac2 93.3%

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

   5. Simplified 93.3%

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

   if -5.59999999999999997e94 < y < 3.6999999999999997e-265 or 1.25000000000000001e-151 < y < 5.2000000000000002e254

   1. Initial program 93.5%

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

      1. associate-*l/ 91.6%

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

      2. associate-/l* 95.6%

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

   3. Simplified 95.6%

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

   if 3.6999999999999997e-265 < y < 1.25000000000000001e-151

   1. Initial program 95.0%

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

      1. associate-*l/ 99.5%

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

      2. associate-/l* 77.1%

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

   3. Simplified 77.1%

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

   5. Taylor expanded in x around inf 99.5%

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

   if 5.2000000000000002e254 < y

   1. Initial program 99.8%

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

      1. associate-*l/ 64.5%

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

      2. associate-/l* 52.9%

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

   3. Simplified 52.9%

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

      1. associate-*r/ 64.5%

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

      2. associate-*l/ 99.8%

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

      3. *-commutative 99.8%

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

      4. clear-num 99.9%

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

      5. un-div-inv 100.0%

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

   6. Applied egg-rr 100.0%

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

   7. Taylor expanded in x around 0 100.0%

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

      1. mul-1-neg 100.0%

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

      2. div-sub 100.0%

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

      3. sub-neg 100.0%

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

      4. *-inverses 100.0%

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

      5. metadata-eval 100.0%

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

   9. Simplified 100.0%

      \[\leadsto \frac{t}{\color{blue}{-\left(\frac{z}{y} + -1\right)}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.6 \cdot 10^{+94}:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-265}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \mathbf{elif}\;y \leq 1.25 \cdot 10^{-151}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+254}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{1 - \frac{z}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \left(1 - \frac{x}{y}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+94}:\\ \;\;\;\;t\_m \cdot \frac{y}{-z}\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{-23} \lor \neg \left(y \leq 1.5 \cdot 10^{-109}\right):\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (- 1.0 (/ x y)))))
   (*
    t_s
    (if (<= y -7e+111)
      t_2
      (if (<= y -5.6e+94)
        (* t_m (/ y (- z)))
        (if (or (<= y -1.26e-23) (not (<= y 1.5e-109)))
          t_2
          (* t_m (/ x z))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (1.0 - (x / y));
	double tmp;
	if (y <= -7e+111) {
		tmp = t_2;
	} else if (y <= -5.6e+94) {
		tmp = t_m * (y / -z);
	} else if ((y <= -1.26e-23) || !(y <= 1.5e-109)) {
		tmp = t_2;
	} else {
		tmp = t_m * (x / z);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (1.0d0 - (x / y))
    if (y <= (-7d+111)) then
        tmp = t_2
    else if (y <= (-5.6d+94)) then
        tmp = t_m * (y / -z)
    else if ((y <= (-1.26d-23)) .or. (.not. (y <= 1.5d-109))) then
        tmp = t_2
    else
        tmp = t_m * (x / z)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (1.0 - (x / y));
	double tmp;
	if (y <= -7e+111) {
		tmp = t_2;
	} else if (y <= -5.6e+94) {
		tmp = t_m * (y / -z);
	} else if ((y <= -1.26e-23) || !(y <= 1.5e-109)) {
		tmp = t_2;
	} else {
		tmp = t_m * (x / z);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = t_m * (1.0 - (x / y))
	tmp = 0
	if y <= -7e+111:
		tmp = t_2
	elif y <= -5.6e+94:
		tmp = t_m * (y / -z)
	elif (y <= -1.26e-23) or not (y <= 1.5e-109):
		tmp = t_2
	else:
		tmp = t_m * (x / z)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m * Float64(1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -7e+111)
		tmp = t_2;
	elseif (y <= -5.6e+94)
		tmp = Float64(t_m * Float64(y / Float64(-z)));
	elseif ((y <= -1.26e-23) || !(y <= 1.5e-109))
		tmp = t_2;
	else
		tmp = Float64(t_m * Float64(x / z));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = t_m * (1.0 - (x / y));
	tmp = 0.0;
	if (y <= -7e+111)
		tmp = t_2;
	elseif (y <= -5.6e+94)
		tmp = t_m * (y / -z);
	elseif ((y <= -1.26e-23) || ~((y <= 1.5e-109)))
		tmp = t_2;
	else
		tmp = t_m * (x / z);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[y, -7e+111], t$95$2, If[LessEqual[y, -5.6e+94], N[(t$95$m * N[(y / (-z)), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -1.26e-23], N[Not[LessEqual[y, 1.5e-109]], $MachinePrecision]], t$95$2, N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if y < -7.0000000000000004e111 or -5.59999999999999997e94 < y < -1.25999999999999996e-23 or 1.50000000000000011e-109 < y

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0 76.7%

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

      1. mul-1-neg 76.7%

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

      2. div-sub 76.7%

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

      3. sub-neg 76.7%

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

      4. *-inverses 76.7%

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

      5. metadata-eval 76.7%

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

   5. Simplified 76.7%

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

   6. Taylor expanded in x around 0 72.6%

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

      1. *-rgt-identity 72.6%

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

      2. *-commutative 72.6%

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

      3. associate-/l* 76.7%

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

      4. associate-*r* 76.7%

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

      5. *-commutative 76.7%

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

      6. distribute-lft-in 76.7%

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

      7. mul-1-neg 76.7%

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

      8. sub-neg 76.7%

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

   8. Simplified 76.7%

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

   if -7.0000000000000004e111 < y < -5.59999999999999997e94

   1. Initial program 100.0%

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

      1. associate-*l/ 83.9%

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

      2. associate-/l* 61.8%

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

   3. Simplified 61.8%

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

   5. Taylor expanded in z around inf 83.9%

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

      1. *-commutative 83.9%

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

      2. associate-/l* 61.8%

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

   7. Simplified 61.8%

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

   8. Taylor expanded in x around 0 68.1%

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

      1. associate-*r/ 68.1%

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

      2. mul-1-neg 68.1%

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

      3. distribute-rgt-neg-out 68.1%

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

      4. associate-/l* 84.0%

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

   10. Simplified 84.0%

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

   if -1.25999999999999996e-23 < y < 1.50000000000000011e-109

   1. Initial program 90.6%

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

   3. Taylor expanded in y around 0 70.6%

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

4. Final simplification 74.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+111}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -5.6 \cdot 10^{+94}:\\ \;\;\;\;t \cdot \frac{y}{-z}\\ \mathbf{elif}\;y \leq -1.26 \cdot 10^{-23} \lor \neg \left(y \leq 1.5 \cdot 10^{-109}\right):\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{x}{z - y}\\ t_3 := t\_m \cdot \left(1 - \frac{x}{y}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+101}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-79}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-279}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (/ x (- z y)))) (t_3 (* t_m (- 1.0 (/ x y)))))
   (*
    t_s
    (if (<= y -5.2e+101)
      t_3
      (if (<= y -2.7e-79)
        t_2
        (if (<= y 8.2e-279)
          (* (- x y) (/ t_m z))
          (if (<= y 6.5e+35) t_2 t_3)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (x / (z - y));
	double t_3 = t_m * (1.0 - (x / y));
	double tmp;
	if (y <= -5.2e+101) {
		tmp = t_3;
	} else if (y <= -2.7e-79) {
		tmp = t_2;
	} else if (y <= 8.2e-279) {
		tmp = (x - y) * (t_m / z);
	} else if (y <= 6.5e+35) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = t_m * (x / (z - y))
    t_3 = t_m * (1.0d0 - (x / y))
    if (y <= (-5.2d+101)) then
        tmp = t_3
    else if (y <= (-2.7d-79)) then
        tmp = t_2
    else if (y <= 8.2d-279) then
        tmp = (x - y) * (t_m / z)
    else if (y <= 6.5d+35) then
        tmp = t_2
    else
        tmp = t_3
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (x / (z - y));
	double t_3 = t_m * (1.0 - (x / y));
	double tmp;
	if (y <= -5.2e+101) {
		tmp = t_3;
	} else if (y <= -2.7e-79) {
		tmp = t_2;
	} else if (y <= 8.2e-279) {
		tmp = (x - y) * (t_m / z);
	} else if (y <= 6.5e+35) {
		tmp = t_2;
	} else {
		tmp = t_3;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = t_m * (x / (z - y))
	t_3 = t_m * (1.0 - (x / y))
	tmp = 0
	if y <= -5.2e+101:
		tmp = t_3
	elif y <= -2.7e-79:
		tmp = t_2
	elif y <= 8.2e-279:
		tmp = (x - y) * (t_m / z)
	elif y <= 6.5e+35:
		tmp = t_2
	else:
		tmp = t_3
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m * Float64(x / Float64(z - y)))
	t_3 = Float64(t_m * Float64(1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -5.2e+101)
		tmp = t_3;
	elseif (y <= -2.7e-79)
		tmp = t_2;
	elseif (y <= 8.2e-279)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (y <= 6.5e+35)
		tmp = t_2;
	else
		tmp = t_3;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = t_m * (x / (z - y));
	t_3 = t_m * (1.0 - (x / y));
	tmp = 0.0;
	if (y <= -5.2e+101)
		tmp = t_3;
	elseif (y <= -2.7e-79)
		tmp = t_2;
	elseif (y <= 8.2e-279)
		tmp = (x - y) * (t_m / z);
	elseif (y <= 6.5e+35)
		tmp = t_2;
	else
		tmp = t_3;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(t$95$m * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[y, -5.2e+101], t$95$3, If[LessEqual[y, -2.7e-79], t$95$2, If[LessEqual[y, 8.2e-279], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.5e+35], t$95$2, t$95$3]]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.2e101 or 6.5000000000000003e35 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 82.1%

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

      1. mul-1-neg 82.1%

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

      2. div-sub 82.1%

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

      3. sub-neg 82.1%

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

      4. *-inverses 82.1%

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

      5. metadata-eval 82.1%

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

   5. Simplified 82.1%

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

   6. Taylor expanded in x around 0 75.1%

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

      1. *-rgt-identity 75.1%

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

      2. *-commutative 75.1%

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

      3. associate-/l* 82.1%

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

      4. associate-*r* 82.1%

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

      5. *-commutative 82.1%

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

      6. distribute-lft-in 82.1%

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

      7. mul-1-neg 82.1%

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

      8. sub-neg 82.1%

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

   8. Simplified 82.1%

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

   if -5.2e101 < y < -2.7000000000000002e-79 or 8.20000000000000034e-279 < y < 6.5000000000000003e35

   1. Initial program 95.8%

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

   3. Taylor expanded in x around inf 73.1%

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

   if -2.7000000000000002e-79 < y < 8.20000000000000034e-279

   1. Initial program 85.8%

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

      1. associate-*l/ 92.7%

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

      2. associate-/l* 98.0%

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

   3. Simplified 98.0%

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

   5. Taylor expanded in z around inf 79.6%

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

      1. *-commutative 79.6%

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

      2. associate-/l* 83.2%

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

   7. Simplified 83.2%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+101}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -2.7 \cdot 10^{-79}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-279}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;y \leq 6.5 \cdot 10^{+35}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \left(1 - \frac{x}{y}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -9 \cdot 10^{+64}:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-265}:\\ \;\;\;\;\frac{x}{\frac{z - y}{t\_m}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+34}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (- 1.0 (/ x y)))))
   (*
    t_s
    (if (<= y -7e+111)
      t_2
      (if (<= y -9e+64)
        (* t_m (/ (- x y) z))
        (if (<= y 1.8e-265)
          (/ x (/ (- z y) t_m))
          (if (<= y 9.2e+34) (/ (* t_m x) (- z y)) t_2)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (1.0 - (x / y));
	double tmp;
	if (y <= -7e+111) {
		tmp = t_2;
	} else if (y <= -9e+64) {
		tmp = t_m * ((x - y) / z);
	} else if (y <= 1.8e-265) {
		tmp = x / ((z - y) / t_m);
	} else if (y <= 9.2e+34) {
		tmp = (t_m * x) / (z - y);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (1.0d0 - (x / y))
    if (y <= (-7d+111)) then
        tmp = t_2
    else if (y <= (-9d+64)) then
        tmp = t_m * ((x - y) / z)
    else if (y <= 1.8d-265) then
        tmp = x / ((z - y) / t_m)
    else if (y <= 9.2d+34) then
        tmp = (t_m * x) / (z - y)
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (1.0 - (x / y));
	double tmp;
	if (y <= -7e+111) {
		tmp = t_2;
	} else if (y <= -9e+64) {
		tmp = t_m * ((x - y) / z);
	} else if (y <= 1.8e-265) {
		tmp = x / ((z - y) / t_m);
	} else if (y <= 9.2e+34) {
		tmp = (t_m * x) / (z - y);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = t_m * (1.0 - (x / y))
	tmp = 0
	if y <= -7e+111:
		tmp = t_2
	elif y <= -9e+64:
		tmp = t_m * ((x - y) / z)
	elif y <= 1.8e-265:
		tmp = x / ((z - y) / t_m)
	elif y <= 9.2e+34:
		tmp = (t_m * x) / (z - y)
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m * Float64(1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -7e+111)
		tmp = t_2;
	elseif (y <= -9e+64)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (y <= 1.8e-265)
		tmp = Float64(x / Float64(Float64(z - y) / t_m));
	elseif (y <= 9.2e+34)
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = t_m * (1.0 - (x / y));
	tmp = 0.0;
	if (y <= -7e+111)
		tmp = t_2;
	elseif (y <= -9e+64)
		tmp = t_m * ((x - y) / z);
	elseif (y <= 1.8e-265)
		tmp = x / ((z - y) / t_m);
	elseif (y <= 9.2e+34)
		tmp = (t_m * x) / (z - y);
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[y, -7e+111], t$95$2, If[LessEqual[y, -9e+64], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.8e-265], N[(x / N[(N[(z - y), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 9.2e+34], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], t$95$2]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if y < -7.0000000000000004e111 or 9.1999999999999993e34 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 83.5%

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

      1. mul-1-neg 83.5%

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

      2. div-sub 83.5%

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

      3. sub-neg 83.5%

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

      4. *-inverses 83.5%

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

      5. metadata-eval 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in x around 0 76.4%

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

      1. *-rgt-identity 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-/l* 83.5%

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

      4. associate-*r* 83.5%

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

      5. *-commutative 83.5%

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

      6. distribute-lft-in 83.5%

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

      7. mul-1-neg 83.5%

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

      8. sub-neg 83.5%

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

   8. Simplified 83.5%

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

   if -7.0000000000000004e111 < y < -8.99999999999999946e64

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 80.8%

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

   if -8.99999999999999946e64 < y < 1.8000000000000001e-265

   1. Initial program 90.9%

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

      1. associate-*l/ 94.0%

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

      2. associate-/l* 97.5%

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

   3. Simplified 97.5%

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

      1. clear-num 96.3%

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

      2. un-div-inv 96.9%

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

   6. Applied egg-rr 96.9%

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

      1. div-sub 94.4%

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

   8. Applied egg-rr 94.4%

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

   9. Taylor expanded in x around inf 78.9%

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

       1. div-sub 81.3%

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

   11. Simplified 81.3%

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

   if 1.8000000000000001e-265 < y < 9.1999999999999993e34

   1. Initial program 93.0%

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

      1. associate-*l/ 96.4%

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

      2. associate-/l* 88.1%

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

   3. Simplified 88.1%

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

   5. Taylor expanded in x around inf 79.7%

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

4. Final simplification 81.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+111}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -9 \cdot 10^{+64}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq 1.8 \cdot 10^{-265}:\\ \;\;\;\;\frac{x}{\frac{z - y}{t}}\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+34}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 60.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+116}:\\ \;\;\;\;t\_m\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;t\_m \cdot \frac{x}{-y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-278}:\\ \;\;\;\;x \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{t\_m}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\_m\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (<= y -4e+116)
    t_m
    (if (<= y -3.6e-22)
      (* t_m (/ x (- y)))
      (if (<= y 2.2e-278)
        (* x (/ t_m z))
        (if (<= y 1.6e+35) (/ t_m (/ z x)) t_m))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (y <= -4e+116) {
		tmp = t_m;
	} else if (y <= -3.6e-22) {
		tmp = t_m * (x / -y);
	} else if (y <= 2.2e-278) {
		tmp = x * (t_m / z);
	} else if (y <= 1.6e+35) {
		tmp = t_m / (z / x);
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (y <= (-4d+116)) then
        tmp = t_m
    else if (y <= (-3.6d-22)) then
        tmp = t_m * (x / -y)
    else if (y <= 2.2d-278) then
        tmp = x * (t_m / z)
    else if (y <= 1.6d+35) then
        tmp = t_m / (z / x)
    else
        tmp = t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (y <= -4e+116) {
		tmp = t_m;
	} else if (y <= -3.6e-22) {
		tmp = t_m * (x / -y);
	} else if (y <= 2.2e-278) {
		tmp = x * (t_m / z);
	} else if (y <= 1.6e+35) {
		tmp = t_m / (z / x);
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if y <= -4e+116:
		tmp = t_m
	elif y <= -3.6e-22:
		tmp = t_m * (x / -y)
	elif y <= 2.2e-278:
		tmp = x * (t_m / z)
	elif y <= 1.6e+35:
		tmp = t_m / (z / x)
	else:
		tmp = t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (y <= -4e+116)
		tmp = t_m;
	elseif (y <= -3.6e-22)
		tmp = Float64(t_m * Float64(x / Float64(-y)));
	elseif (y <= 2.2e-278)
		tmp = Float64(x * Float64(t_m / z));
	elseif (y <= 1.6e+35)
		tmp = Float64(t_m / Float64(z / x));
	else
		tmp = t_m;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (y <= -4e+116)
		tmp = t_m;
	elseif (y <= -3.6e-22)
		tmp = t_m * (x / -y);
	elseif (y <= 2.2e-278)
		tmp = x * (t_m / z);
	elseif (y <= 1.6e+35)
		tmp = t_m / (z / x);
	else
		tmp = t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[y, -4e+116], t$95$m, If[LessEqual[y, -3.6e-22], N[(t$95$m * N[(x / (-y)), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.2e-278], N[(x * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.6e+35], N[(t$95$m / N[(z / x), $MachinePrecision]), $MachinePrecision], t$95$m]]]]), $MachinePrecision]

Derivation

1. Split input into 4 regimes

2. if y < -4.00000000000000006e116 or 1.59999999999999991e35 < y

   1. Initial program 99.8%

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

      1. associate-*l/ 69.4%

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

      2. associate-/l* 73.1%

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

   3. Simplified 73.1%

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

   5. Taylor expanded in y around inf 70.7%

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

   if -4.00000000000000006e116 < y < -3.5999999999999998e-22

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 54.4%

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

      1. mul-1-neg 54.4%

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

      2. div-sub 54.4%

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

      3. sub-neg 54.4%

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

      4. *-inverses 54.4%

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

      5. metadata-eval 54.4%

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

   5. Simplified 54.4%

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

   6. Taylor expanded in x around 0 54.4%

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

      1. *-rgt-identity 54.4%

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

      2. *-commutative 54.4%

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

      3. associate-/l* 54.4%

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

      4. associate-*r* 54.4%

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

      5. *-commutative 54.4%

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

      6. distribute-lft-in 54.4%

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

      7. mul-1-neg 54.4%

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

      8. sub-neg 54.4%

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

   8. Simplified 54.4%

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

   9. Taylor expanded in x around inf 48.0%

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

       1. mul-1-neg 48.0%

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

       2. associate-/l* 48.0%

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

       3. distribute-rgt-neg-in 48.0%

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

       4. distribute-neg-frac 48.0%

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

   11. Simplified 48.0%

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

   if -3.5999999999999998e-22 < y < 2.2000000000000001e-278

   1. Initial program 88.6%

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

      1. associate-*l/ 92.7%

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

      2. associate-/l* 98.4%

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

   3. Simplified 98.4%

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

   5. Taylor expanded in y around 0 65.1%

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

      1. *-commutative 65.1%

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

      2. associate-/l* 70.7%

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

   7. Simplified 70.7%

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

   if 2.2000000000000001e-278 < y < 1.59999999999999991e35

   1. Initial program 93.5%

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

      1. associate-*l/ 96.5%

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

      2. associate-/l* 87.3%

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

   3. Simplified 87.3%

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

      1. associate-*r/ 96.5%

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

      2. associate-*l/ 93.5%

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

      3. *-commutative 93.5%

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

      4. clear-num 93.5%

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

      5. un-div-inv 94.2%

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

   6. Applied egg-rr 94.2%

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

   7. Taylor expanded in y around 0 63.3%

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

4. Final simplification 66.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+116}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq -3.6 \cdot 10^{-22}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \mathbf{elif}\;y \leq 2.2 \cdot 10^{-278}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+35}:\\ \;\;\;\;\frac{t}{\frac{z}{x}}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \left(1 - \frac{x}{y}\right)\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+111}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{+66}:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{\frac{z - y}{t\_m}}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (- 1.0 (/ x y)))))
   (*
    t_s
    (if (<= y -7e+111)
      t_2
      (if (<= y -4.3e+66)
        (* t_m (/ (- x y) z))
        (if (<= y 7.2e+41) (/ x (/ (- z y) t_m)) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (1.0 - (x / y));
	double tmp;
	if (y <= -7e+111) {
		tmp = t_2;
	} else if (y <= -4.3e+66) {
		tmp = t_m * ((x - y) / z);
	} else if (y <= 7.2e+41) {
		tmp = x / ((z - y) / t_m);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = t_m * (1.0d0 - (x / y))
    if (y <= (-7d+111)) then
        tmp = t_2
    else if (y <= (-4.3d+66)) then
        tmp = t_m * ((x - y) / z)
    else if (y <= 7.2d+41) then
        tmp = x / ((z - y) / t_m)
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (1.0 - (x / y));
	double tmp;
	if (y <= -7e+111) {
		tmp = t_2;
	} else if (y <= -4.3e+66) {
		tmp = t_m * ((x - y) / z);
	} else if (y <= 7.2e+41) {
		tmp = x / ((z - y) / t_m);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = t_m * (1.0 - (x / y))
	tmp = 0
	if y <= -7e+111:
		tmp = t_2
	elif y <= -4.3e+66:
		tmp = t_m * ((x - y) / z)
	elif y <= 7.2e+41:
		tmp = x / ((z - y) / t_m)
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m * Float64(1.0 - Float64(x / y)))
	tmp = 0.0
	if (y <= -7e+111)
		tmp = t_2;
	elseif (y <= -4.3e+66)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (y <= 7.2e+41)
		tmp = Float64(x / Float64(Float64(z - y) / t_m));
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = t_m * (1.0 - (x / y));
	tmp = 0.0;
	if (y <= -7e+111)
		tmp = t_2;
	elseif (y <= -4.3e+66)
		tmp = t_m * ((x - y) / z);
	elseif (y <= 7.2e+41)
		tmp = x / ((z - y) / t_m);
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[y, -7e+111], t$95$2, If[LessEqual[y, -4.3e+66], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 7.2e+41], N[(x / N[(N[(z - y), $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision], t$95$2]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if y < -7.0000000000000004e111 or 7.20000000000000051e41 < y

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0 83.5%

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

      1. mul-1-neg 83.5%

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

      2. div-sub 83.5%

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

      3. sub-neg 83.5%

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

      4. *-inverses 83.5%

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

      5. metadata-eval 83.5%

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

   5. Simplified 83.5%

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

   6. Taylor expanded in x around 0 76.4%

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

      1. *-rgt-identity 76.4%

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

      2. *-commutative 76.4%

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

      3. associate-/l* 83.5%

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

      4. associate-*r* 83.5%

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

      5. *-commutative 83.5%

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

      6. distribute-lft-in 83.5%

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

      7. mul-1-neg 83.5%

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

      8. sub-neg 83.5%

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

   8. Simplified 83.5%

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

   if -7.0000000000000004e111 < y < -4.30000000000000027e66

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf 80.8%

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

   if -4.30000000000000027e66 < y < 7.20000000000000051e41

   1. Initial program 91.8%

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

      1. associate-*l/ 95.0%

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

      2. associate-/l* 93.6%

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

   3. Simplified 93.6%

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

      1. clear-num 92.8%

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

      2. un-div-inv 93.8%

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

   6. Applied egg-rr 93.8%

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

      1. div-sub 92.3%

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

   8. Applied egg-rr 92.3%

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

   9. Taylor expanded in x around inf 76.3%

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

       1. div-sub 77.8%

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

   11. Simplified 77.8%

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

4. Final simplification 80.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7 \cdot 10^{+111}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \mathbf{elif}\;y \leq -4.3 \cdot 10^{+66}:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;y \leq 7.2 \cdot 10^{+41}:\\ \;\;\;\;\frac{x}{\frac{z - y}{t}}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.5% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-29} \lor \neg \left(z \leq 2.7 \cdot 10^{+81}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (or (<= z -3.3e-29) (not (<= z 2.7e+81)))
    (* (- x y) (/ t_m z))
    (* t_m (- 1.0 (/ x y))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if ((z <= -3.3e-29) || !(z <= 2.7e+81)) {
		tmp = (x - y) * (t_m / z);
	} else {
		tmp = t_m * (1.0 - (x / y));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if ((z <= (-3.3d-29)) .or. (.not. (z <= 2.7d+81))) then
        tmp = (x - y) * (t_m / z)
    else
        tmp = t_m * (1.0d0 - (x / y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if ((z <= -3.3e-29) || !(z <= 2.7e+81)) {
		tmp = (x - y) * (t_m / z);
	} else {
		tmp = t_m * (1.0 - (x / y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if (z <= -3.3e-29) or not (z <= 2.7e+81):
		tmp = (x - y) * (t_m / z)
	else:
		tmp = t_m * (1.0 - (x / y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if ((z <= -3.3e-29) || !(z <= 2.7e+81))
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	else
		tmp = Float64(t_m * Float64(1.0 - Float64(x / y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if ((z <= -3.3e-29) || ~((z <= 2.7e+81)))
		tmp = (x - y) * (t_m / z);
	else
		tmp = t_m * (1.0 - (x / y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[Or[LessEqual[z, -3.3e-29], N[Not[LessEqual[z, 2.7e+81]], $MachinePrecision]], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -3.30000000000000028e-29 or 2.6999999999999999e81 < z

   1. Initial program 96.7%

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

      1. associate-*l/ 86.6%

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

      2. associate-/l* 84.4%

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

   3. Simplified 84.4%

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

   5. Taylor expanded in z around inf 70.9%

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

      1. *-commutative 70.9%

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

      2. associate-/l* 68.1%

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

   7. Simplified 68.1%

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

   if -3.30000000000000028e-29 < z < 2.6999999999999999e81

   1. Initial program 94.5%

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

   3. Taylor expanded in z around 0 75.9%

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

      1. mul-1-neg 75.9%

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

      2. div-sub 76.0%

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

      3. sub-neg 76.0%

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

      4. *-inverses 76.0%

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

      5. metadata-eval 76.0%

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

   5. Simplified 76.0%

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

   6. Taylor expanded in x around 0 76.2%

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

      1. *-rgt-identity 76.2%

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

      2. *-commutative 76.2%

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

      3. associate-/l* 76.0%

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

      4. associate-*r* 76.0%

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

      5. *-commutative 76.0%

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

      6. distribute-lft-in 76.0%

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

      7. mul-1-neg 76.0%

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

      8. sub-neg 76.0%

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

   8. Simplified 76.0%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-29} \lor \neg \left(z \leq 2.7 \cdot 10^{+81}\right):\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+46}:\\ \;\;\;\;t\_m \cdot \frac{y}{y - z}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+39}:\\ \;\;\;\;\frac{t\_m \cdot x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (<= y -6.2e+46)
    (* t_m (/ y (- y z)))
    (if (<= y 1.85e+39) (/ (* t_m x) (- z y)) (* t_m (- 1.0 (/ x y)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (y <= -6.2e+46) {
		tmp = t_m * (y / (y - z));
	} else if (y <= 1.85e+39) {
		tmp = (t_m * x) / (z - y);
	} else {
		tmp = t_m * (1.0 - (x / y));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (y <= (-6.2d+46)) then
        tmp = t_m * (y / (y - z))
    else if (y <= 1.85d+39) then
        tmp = (t_m * x) / (z - y)
    else
        tmp = t_m * (1.0d0 - (x / y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (y <= -6.2e+46) {
		tmp = t_m * (y / (y - z));
	} else if (y <= 1.85e+39) {
		tmp = (t_m * x) / (z - y);
	} else {
		tmp = t_m * (1.0 - (x / y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if y <= -6.2e+46:
		tmp = t_m * (y / (y - z))
	elif y <= 1.85e+39:
		tmp = (t_m * x) / (z - y)
	else:
		tmp = t_m * (1.0 - (x / y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (y <= -6.2e+46)
		tmp = Float64(t_m * Float64(y / Float64(y - z)));
	elseif (y <= 1.85e+39)
		tmp = Float64(Float64(t_m * x) / Float64(z - y));
	else
		tmp = Float64(t_m * Float64(1.0 - Float64(x / y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (y <= -6.2e+46)
		tmp = t_m * (y / (y - z));
	elseif (y <= 1.85e+39)
		tmp = (t_m * x) / (z - y);
	else
		tmp = t_m * (1.0 - (x / y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[y, -6.2e+46], N[(t$95$m * N[(y / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.85e+39], N[(N[(t$95$m * x), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(t$95$m * N[(1.0 - N[(x / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -6.1999999999999995e46

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0 89.7%

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

      1. neg-mul-1 89.7%

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

      2. distribute-neg-frac2 89.7%

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

   5. Simplified 89.7%

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

   if -6.1999999999999995e46 < y < 1.85000000000000006e39

   1. Initial program 91.7%

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

      1. associate-*l/ 95.0%

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

      2. associate-/l* 93.5%

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

   3. Simplified 93.5%

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

   5. Taylor expanded in x around inf 78.3%

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

   if 1.85000000000000006e39 < y

   1. Initial program 99.7%

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

   3. Taylor expanded in z around 0 77.3%

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

      1. mul-1-neg 77.3%

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

      2. div-sub 77.4%

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

      3. sub-neg 77.4%

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

      4. *-inverses 77.4%

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

      5. metadata-eval 77.4%

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

   5. Simplified 77.4%

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

   6. Taylor expanded in x around 0 72.2%

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

      1. *-rgt-identity 72.2%

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

      2. *-commutative 72.2%

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

      3. associate-/l* 77.4%

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

      4. associate-*r* 77.4%

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

      5. *-commutative 77.4%

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

      6. distribute-lft-in 77.4%

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

      7. mul-1-neg 77.4%

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

      8. sub-neg 77.4%

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

   8. Simplified 77.4%

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

4. Final simplification 81.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{+46}:\\ \;\;\;\;t \cdot \frac{y}{y - z}\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{+39}:\\ \;\;\;\;\frac{t \cdot x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(1 - \frac{x}{y}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+101}:\\ \;\;\;\;t\_m\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \frac{t\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_m\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (* t_s (if (<= y -5.2e+101) t_m (if (<= y 3e+35) (* x (/ t_m z)) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (y <= -5.2e+101) {
		tmp = t_m;
	} else if (y <= 3e+35) {
		tmp = x * (t_m / z);
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (y <= (-5.2d+101)) then
        tmp = t_m
    else if (y <= 3d+35) then
        tmp = x * (t_m / z)
    else
        tmp = t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (y <= -5.2e+101) {
		tmp = t_m;
	} else if (y <= 3e+35) {
		tmp = x * (t_m / z);
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if y <= -5.2e+101:
		tmp = t_m
	elif y <= 3e+35:
		tmp = x * (t_m / z)
	else:
		tmp = t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (y <= -5.2e+101)
		tmp = t_m;
	elseif (y <= 3e+35)
		tmp = Float64(x * Float64(t_m / z));
	else
		tmp = t_m;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (y <= -5.2e+101)
		tmp = t_m;
	elseif (y <= 3e+35)
		tmp = x * (t_m / z);
	else
		tmp = t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[y, -5.2e+101], t$95$m, If[LessEqual[y, 3e+35], N[(x * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], t$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -5.2e101 or 2.99999999999999991e35 < y

   1. Initial program 99.8%

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

      1. associate-*l/ 69.3%

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

      2. associate-/l* 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in y around inf 68.8%

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

   if -5.2e101 < y < 2.99999999999999991e35

   1. Initial program 92.3%

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

      1. associate-*l/ 95.3%

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

      2. associate-/l* 93.1%

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

   3. Simplified 93.1%

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

   5. Taylor expanded in y around 0 59.4%

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

      1. *-commutative 59.4%

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

      2. associate-/l* 58.0%

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

   7. Simplified 58.0%

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

4. Final simplification 62.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+101}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 3 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 60.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+101}:\\ \;\;\;\;t\_m\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+44}:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_m\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (* t_s (if (<= y -5.2e+101) t_m (if (<= y 4.7e+44) (* t_m (/ x z)) t_m))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (y <= -5.2e+101) {
		tmp = t_m;
	} else if (y <= 4.7e+44) {
		tmp = t_m * (x / z);
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (y <= (-5.2d+101)) then
        tmp = t_m
    else if (y <= 4.7d+44) then
        tmp = t_m * (x / z)
    else
        tmp = t_m
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (y <= -5.2e+101) {
		tmp = t_m;
	} else if (y <= 4.7e+44) {
		tmp = t_m * (x / z);
	} else {
		tmp = t_m;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if y <= -5.2e+101:
		tmp = t_m
	elif y <= 4.7e+44:
		tmp = t_m * (x / z)
	else:
		tmp = t_m
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (y <= -5.2e+101)
		tmp = t_m;
	elseif (y <= 4.7e+44)
		tmp = Float64(t_m * Float64(x / z));
	else
		tmp = t_m;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (y <= -5.2e+101)
		tmp = t_m;
	elseif (y <= 4.7e+44)
		tmp = t_m * (x / z);
	else
		tmp = t_m;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[y, -5.2e+101], t$95$m, If[LessEqual[y, 4.7e+44], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$m]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -5.2e101 or 4.70000000000000018e44 < y

   1. Initial program 99.8%

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

      1. associate-*l/ 69.3%

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

      2. associate-/l* 73.0%

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

   3. Simplified 73.0%

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

   5. Taylor expanded in y around inf 68.8%

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

   if -5.2e101 < y < 4.70000000000000018e44

   1. Initial program 92.3%

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

   3. Taylor expanded in y around 0 60.6%

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

4. Final simplification 64.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.2 \cdot 10^{+101}:\\ \;\;\;\;t\\ \mathbf{elif}\;y \leq 4.7 \cdot 10^{+44}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 36.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{+228}:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (* t_s (if (<= z 5.4e+228) t_m (* t_m (/ y z)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (z <= 5.4e+228) {
		tmp = t_m;
	} else {
		tmp = t_m * (y / z);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (z <= 5.4d+228) then
        tmp = t_m
    else
        tmp = t_m * (y / z)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (z <= 5.4e+228) {
		tmp = t_m;
	} else {
		tmp = t_m * (y / z);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if z <= 5.4e+228:
		tmp = t_m
	else:
		tmp = t_m * (y / z)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (z <= 5.4e+228)
		tmp = t_m;
	else
		tmp = Float64(t_m * Float64(y / z));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (z <= 5.4e+228)
		tmp = t_m;
	else
		tmp = t_m * (y / z);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[z, 5.4e+228], t$95$m, N[(t$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 5.4000000000000003e228

   1. Initial program 95.7%

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

      1. associate-*l/ 83.6%

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

      2. associate-/l* 84.3%

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

   3. Simplified 84.3%

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

   5. Taylor expanded in y around inf 40.6%

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

   if 5.4000000000000003e228 < z

   1. Initial program 92.9%

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

      1. associate-*l/ 91.6%

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

      2. associate-/l* 87.6%

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

   3. Simplified 87.6%

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

   5. Taylor expanded in z around inf 85.6%

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

      1. *-commutative 85.6%

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

      2. associate-/l* 87.6%

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

   7. Simplified 87.6%

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

   8. Taylor expanded in x around 0 61.1%

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

      1. associate-*r/ 61.1%

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

      2. mul-1-neg 61.1%

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

      3. distribute-rgt-neg-out 61.1%

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

      4. associate-/l* 69.3%

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

   10. Simplified 69.3%

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

       1. clear-num 69.3%

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

       2. un-div-inv 69.3%

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

       3. add-sqr-sqrt 30.8%

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

       4. sqrt-unprod 53.7%

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

       5. sqr-neg 53.7%

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

       6. sqrt-unprod 23.3%

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

       7. add-sqr-sqrt 49.9%

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

   12. Applied egg-rr 49.9%

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

       1. associate-/r/ 50.0%

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

       2. associate-*l/ 49.8%

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

       3. associate-/l* 49.9%

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

   14. Simplified 49.9%

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

4. Final simplification 41.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.4 \cdot 10^{+228}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 97.2% accurate, 1.0× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (* t_s (* t_m (/ (- x y) (- z y)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * (t_m * ((x - y) / (z - y)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = t_s * (t_m * ((x - y) / (z - y)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * (t_m * ((x - y) / (z - y)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	return t_s * (t_m * ((x - y) / (z - y)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	return Float64(t_s * Float64(t_m * Float64(Float64(x - y) / Float64(z - y))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, y, z, t_m)
	tmp = t_s * (t_m * ((x - y) / (z - y)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * N[(t$95$m * N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

3. Final simplification 95.5%

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

Alternative 16: 97.2% accurate, 1.0× speedup

Math

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

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (* t_s (/ t_m (/ (- z y) (- x y)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * (t_m / ((z - y) / (x - y)));
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = t_s * (t_m / ((z - y) / (x - y)))
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * (t_m / ((z - y) / (x - y)));
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	return t_s * (t_m / ((z - y) / (x - y)))

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	return Float64(t_s * Float64(t_m / Float64(Float64(z - y) / Float64(x - y))))
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, y, z, t_m)
	tmp = t_s * (t_m / ((z - y) / (x - y)));
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * N[(t$95$m / N[(N[(z - y), $MachinePrecision] / N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

   1. associate-*l/ 84.3%

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

   2. associate-/l* 84.6%

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

3. Simplified 84.6%

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

   1. associate-*r/ 84.3%

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

   2. associate-*l/ 95.5%

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

   3. *-commutative 95.5%

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

   4. clear-num 95.0%

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

   5. un-div-inv 95.8%

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

6. Applied egg-rr 95.8%

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

7. Final simplification 95.8%

   \[\leadsto \frac{t}{\frac{z - y}{x - y}} \]
8. Add Preprocessing

Alternative 17: 35.1% accurate, 9.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot t\_m \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 1 t)
(FPCore (t_s x y z t_m) :precision binary64 (* t_s t_m))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * t_m;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = t_s * t_m
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * t_m;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	return t_s * t_m

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	return Float64(t_s * t_m)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, y, z, t_m)
	tmp = t_s * t_m;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * t$95$m), $MachinePrecision]

Derivation

1. Initial program 95.5%

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

   1. associate-*l/ 84.3%

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

   2. associate-/l* 84.6%

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

3. Simplified 84.6%

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

5. Taylor expanded in y around inf 37.4%

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

6. Final simplification 37.4%

   \[\leadsto t \]
7. Add Preprocessing

Developer target: 97.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{t}{\frac{z - y}{x - y}} \end{array} \]

FPCore

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

C

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

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 = t / ((z - y) / (x - y))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024096 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

  :alt
  (/ t (/ (- z y) (- x y)))

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