Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C

Percentage Accurate: 94.8% → 95.6%

Time: 10.6s

Alternatives: 11

Speedup: 0.6×

• 21.4% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\begin{array}{l} \\ x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))

C

double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

Fortran

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

Java

public static double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

Python

def code(x, y, z, t):
	return x * ((y / z) - (t / (1.0 - z)))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * ((y / z) - (t / (1.0 - z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 94.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x (- (/ y z) (/ t (- 1.0 z)))))

C

double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

Fortran

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

Java

public static double code(double x, double y, double z, double t) {
	return x * ((y / z) - (t / (1.0 - z)));
}

Python

def code(x, y, z, t):
	return x * ((y / z) - (t / (1.0 - z)))

Julia

function code(x, y, z, t)
	return Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * ((y / z) - (t / (1.0 - z)));
end

Wolfram

code[x_, y_, z_, t_] := N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 95.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{t}{1 - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y - x\_m \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\_1\right)\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (/ t (- 1.0 z))))
   (*
    x_s
    (if (<= x_m 1.5e-12)
      (- (* (/ x_m z) y) (* x_m t_1))
      (* x_m (- (/ y z) t_1))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = t / (1.0 - z);
	double tmp;
	if (x_m <= 1.5e-12) {
		tmp = ((x_m / z) * y) - (x_m * t_1);
	} else {
		tmp = x_m * ((y / z) - t_1);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t / (1.0d0 - z)
    if (x_m <= 1.5d-12) then
        tmp = ((x_m / z) * y) - (x_m * t_1)
    else
        tmp = x_m * ((y / z) - t_1)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = t / (1.0 - z);
	double tmp;
	if (x_m <= 1.5e-12) {
		tmp = ((x_m / z) * y) - (x_m * t_1);
	} else {
		tmp = x_m * ((y / z) - t_1);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = t / (1.0 - z)
	tmp = 0
	if x_m <= 1.5e-12:
		tmp = ((x_m / z) * y) - (x_m * t_1)
	else:
		tmp = x_m * ((y / z) - t_1)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(t / Float64(1.0 - z))
	tmp = 0.0
	if (x_m <= 1.5e-12)
		tmp = Float64(Float64(Float64(x_m / z) * y) - Float64(x_m * t_1));
	else
		tmp = Float64(x_m * Float64(Float64(y / z) - t_1));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = t / (1.0 - z);
	tmp = 0.0;
	if (x_m <= 1.5e-12)
		tmp = ((x_m / z) * y) - (x_m * t_1);
	else
		tmp = x_m * ((y / z) - t_1);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[x$95$m, 1.5e-12], N[(N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision] - N[(x$95$m * t$95$1), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t$95$1), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.5000000000000001e-12

   1. Initial program 94.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 94.2%

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

      2. associate-/r/ 94.2%

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in y around 0 88.5%

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

      1. +-commutative 88.5%

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

      2. mul-1-neg 88.5%

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

      3. associate-/l* 89.6%

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

      4. unsub-neg 89.6%

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

      5. associate-/l* 89.7%

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

      6. associate-/r/ 92.8%

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

      7. associate-/r/ 94.5%

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

      8. *-commutative 94.5%

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

   7. Simplified 94.5%

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

   if 1.5000000000000001e-12 < x

   1. Initial program 99.7%

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

4. Final simplification 96.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.5 \cdot 10^{-12}:\\ \;\;\;\;\frac{x}{z} \cdot y - x \cdot \frac{t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 74.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+23}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 620000000:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+59}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+105}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -1.85e+23)
    (* x_m (/ y z))
    (if (<= z 620000000.0)
      (* x_m (- (/ y z) t))
      (if (<= z 1.02e+59)
        (/ x_m (/ z t))
        (if (<= z 1.75e+105) (* (/ x_m z) y) (* x_m (/ t z))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.85e+23) {
		tmp = x_m * (y / z);
	} else if (z <= 620000000.0) {
		tmp = x_m * ((y / z) - t);
	} else if (z <= 1.02e+59) {
		tmp = x_m / (z / t);
	} else if (z <= 1.75e+105) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m * (t / z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.85d+23)) then
        tmp = x_m * (y / z)
    else if (z <= 620000000.0d0) then
        tmp = x_m * ((y / z) - t)
    else if (z <= 1.02d+59) then
        tmp = x_m / (z / t)
    else if (z <= 1.75d+105) then
        tmp = (x_m / z) * y
    else
        tmp = x_m * (t / z)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -1.85e+23) {
		tmp = x_m * (y / z);
	} else if (z <= 620000000.0) {
		tmp = x_m * ((y / z) - t);
	} else if (z <= 1.02e+59) {
		tmp = x_m / (z / t);
	} else if (z <= 1.75e+105) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m * (t / z);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -1.85e+23:
		tmp = x_m * (y / z)
	elif z <= 620000000.0:
		tmp = x_m * ((y / z) - t)
	elif z <= 1.02e+59:
		tmp = x_m / (z / t)
	elif z <= 1.75e+105:
		tmp = (x_m / z) * y
	else:
		tmp = x_m * (t / z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -1.85e+23)
		tmp = Float64(x_m * Float64(y / z));
	elseif (z <= 620000000.0)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	elseif (z <= 1.02e+59)
		tmp = Float64(x_m / Float64(z / t));
	elseif (z <= 1.75e+105)
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = Float64(x_m * Float64(t / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -1.85e+23)
		tmp = x_m * (y / z);
	elseif (z <= 620000000.0)
		tmp = x_m * ((y / z) - t);
	elseif (z <= 1.02e+59)
		tmp = x_m / (z / t);
	elseif (z <= 1.75e+105)
		tmp = (x_m / z) * y;
	else
		tmp = x_m * (t / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -1.85e+23], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 620000000.0], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.02e+59], N[(x$95$m / N[(z / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.75e+105], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if z < -1.85000000000000006e23

   1. Initial program 97.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 63.2%

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

      1. associate-*r/ 69.0%

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

   5. Simplified 69.0%

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

   if -1.85000000000000006e23 < z < 6.2e8

   1. Initial program 94.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 91.6%

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

      1. +-commutative 91.6%

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

      2. associate-*r/ 88.5%

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

      3. *-commutative 88.5%

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

      4. associate-*r* 88.5%

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

      5. neg-mul-1 88.5%

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

      6. distribute-rgt-out 92.5%

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

      7. unsub-neg 92.5%

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

   5. Simplified 92.5%

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

   if 6.2e8 < z < 1.02000000000000002e59

   1. Initial program 100.0%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 100.0%

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

      2. associate-/r/ 99.7%

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

   4. Applied egg-rr 99.7%

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

   5. Taylor expanded in z around inf 95.2%

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

      1. associate-/l* 95.4%

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

      2. sub-neg 95.4%

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

      3. mul-1-neg 95.4%

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

      4. remove-double-neg 95.4%

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

   7. Simplified 95.4%

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

   8. Taylor expanded in y around 0 80.1%

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

   if 1.02000000000000002e59 < z < 1.74999999999999996e105

   1. Initial program 99.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 76.0%

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

      1. associate-/l* 83.6%

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

      2. associate-/r/ 84.1%

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

   5. Simplified 84.1%

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

   if 1.74999999999999996e105 < z

   1. Initial program 98.3%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 98.2%

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

      2. associate-/r/ 98.2%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in z around inf 81.8%

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

      1. associate-/l* 98.2%

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

      2. sub-neg 98.2%

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

      3. mul-1-neg 98.2%

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

      4. remove-double-neg 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in y around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. associate-*r/ 70.8%

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

   10. Simplified 70.8%

       \[\leadsto \color{blue}{x \cdot \frac{t}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 83.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+23}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 620000000:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 1.02 \cdot 10^{+59}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \mathbf{elif}\;z \leq 1.75 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 74.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+34}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 920000000:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+58}:\\ \;\;\;\;x\_m \cdot \frac{t}{z + -1}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -5.2e+34)
    (* x_m (/ y z))
    (if (<= z 920000000.0)
      (* x_m (- (/ y z) t))
      (if (<= z 2.3e+58)
        (* x_m (/ t (+ z -1.0)))
        (if (<= z 3.8e+105) (* (/ x_m z) y) (* x_m (/ t z))))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -5.2e+34) {
		tmp = x_m * (y / z);
	} else if (z <= 920000000.0) {
		tmp = x_m * ((y / z) - t);
	} else if (z <= 2.3e+58) {
		tmp = x_m * (t / (z + -1.0));
	} else if (z <= 3.8e+105) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m * (t / z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-5.2d+34)) then
        tmp = x_m * (y / z)
    else if (z <= 920000000.0d0) then
        tmp = x_m * ((y / z) - t)
    else if (z <= 2.3d+58) then
        tmp = x_m * (t / (z + (-1.0d0)))
    else if (z <= 3.8d+105) then
        tmp = (x_m / z) * y
    else
        tmp = x_m * (t / z)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -5.2e+34) {
		tmp = x_m * (y / z);
	} else if (z <= 920000000.0) {
		tmp = x_m * ((y / z) - t);
	} else if (z <= 2.3e+58) {
		tmp = x_m * (t / (z + -1.0));
	} else if (z <= 3.8e+105) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m * (t / z);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -5.2e+34:
		tmp = x_m * (y / z)
	elif z <= 920000000.0:
		tmp = x_m * ((y / z) - t)
	elif z <= 2.3e+58:
		tmp = x_m * (t / (z + -1.0))
	elif z <= 3.8e+105:
		tmp = (x_m / z) * y
	else:
		tmp = x_m * (t / z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -5.2e+34)
		tmp = Float64(x_m * Float64(y / z));
	elseif (z <= 920000000.0)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	elseif (z <= 2.3e+58)
		tmp = Float64(x_m * Float64(t / Float64(z + -1.0)));
	elseif (z <= 3.8e+105)
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = Float64(x_m * Float64(t / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -5.2e+34)
		tmp = x_m * (y / z);
	elseif (z <= 920000000.0)
		tmp = x_m * ((y / z) - t);
	elseif (z <= 2.3e+58)
		tmp = x_m * (t / (z + -1.0));
	elseif (z <= 3.8e+105)
		tmp = (x_m / z) * y;
	else
		tmp = x_m * (t / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -5.2e+34], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 920000000.0], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.3e+58], N[(x$95$m * N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.8e+105], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]

Derivation

1. Split input into 5 regimes

2. if z < -5.19999999999999995e34

   1. Initial program 97.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 63.2%

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

      1. associate-*r/ 69.0%

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

   5. Simplified 69.0%

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

   if -5.19999999999999995e34 < z < 9.2e8

   1. Initial program 94.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 91.6%

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

      1. +-commutative 91.6%

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

      2. associate-*r/ 88.5%

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

      3. *-commutative 88.5%

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

      4. associate-*r* 88.5%

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

      5. neg-mul-1 88.5%

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

      6. distribute-rgt-out 92.5%

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

      7. unsub-neg 92.5%

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

   5. Simplified 92.5%

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

   if 9.2e8 < z < 2.30000000000000002e58

   1. Initial program 100.0%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0 84.4%

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

      1. associate-*r/ 84.4%

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

      2. associate-*r* 84.4%

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

      3. neg-mul-1 84.4%

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

      4. associate-*l/ 84.7%

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

      5. *-commutative 84.7%

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

      6. distribute-frac-neg 84.7%

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

      7. mul-1-neg 84.7%

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

      8. *-commutative 84.7%

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

      9. associate-*l/ 84.7%

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

      10. associate-*r/ 84.4%

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

      11. metadata-eval 84.4%

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

      12. associate-/r* 84.4%

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

      13. neg-mul-1 84.4%

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

      14. associate-*r/ 84.7%

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

      15. *-rgt-identity 84.7%

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

      16. neg-sub0 84.7%

          \[\leadsto x \cdot \frac{t}{\color{blue}{0 - \left(1 - z\right)}} \]

      17. associate--r- 84.7%

          \[\leadsto x \cdot \frac{t}{\color{blue}{\left(0 - 1\right) + z}} \]

      18. metadata-eval 84.7%

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

   5. Simplified 84.7%

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

   if 2.30000000000000002e58 < z < 3.8e105

   1. Initial program 99.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 76.0%

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

      1. associate-/l* 83.6%

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

      2. associate-/r/ 84.1%

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

   5. Simplified 84.1%

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

   if 3.8e105 < z

   1. Initial program 98.3%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 98.2%

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

      2. associate-/r/ 98.2%

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

   4. Applied egg-rr 98.2%

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

   5. Taylor expanded in z around inf 81.8%

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

      1. associate-/l* 98.2%

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

      2. sub-neg 98.2%

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

      3. mul-1-neg 98.2%

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

      4. remove-double-neg 98.2%

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

   7. Simplified 98.2%

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

   8. Taylor expanded in y around 0 61.0%

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

      1. *-commutative 61.0%

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

      2. associate-*r/ 70.8%

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

   10. Simplified 70.8%

       \[\leadsto \color{blue}{x \cdot \frac{t}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.2 \cdot 10^{+34}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 920000000:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 2.3 \cdot 10^{+58}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{elif}\;z \leq 3.8 \cdot 10^{+105}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t\_1 \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x\_m \cdot t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \end{array} \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))))
   (* x_s (if (<= t_1 5e+306) (* x_m t_1) (* (/ x_m z) y)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= 5e+306) {
		tmp = x_m * t_1;
	} else {
		tmp = (x_m / z) * y;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y / z) - (t / (1.0d0 - z))
    if (t_1 <= 5d+306) then
        tmp = x_m * t_1
    else
        tmp = (x_m / z) * y
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_1 <= 5e+306) {
		tmp = x_m * t_1;
	} else {
		tmp = (x_m / z) * y;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_1 <= 5e+306:
		tmp = x_m * t_1
	else:
		tmp = (x_m / z) * y
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_1 <= 5e+306)
		tmp = Float64(x_m * t_1);
	else
		tmp = Float64(Float64(x_m / z) * y);
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_1 <= 5e+306)
		tmp = x_m * t_1;
	else
		tmp = (x_m / z) * y;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, 5e+306], N[(x$95$m * t$95$1), $MachinePrecision], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 4.99999999999999993e306

   1. Initial program 97.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   if 4.99999999999999993e306 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 79.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 99.9%

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

      1. associate-/l* 80.3%

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

      2. associate-/r/ 99.9%

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

   5. Simplified 99.9%

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

4. Final simplification 97.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq 5 \cdot 10^{+306}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9 \lor \neg \left(z \leq 0.0005\right):\\ \;\;\;\;x\_m \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -9.0) (not (<= z 0.0005)))
    (* x_m (/ (+ y t) z))
    (* x_m (- (/ y z) t)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -9.0) || !(z <= 0.0005)) {
		tmp = x_m * ((y + t) / z);
	} else {
		tmp = x_m * ((y / z) - t);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-9.0d0)) .or. (.not. (z <= 0.0005d0))) then
        tmp = x_m * ((y + t) / z)
    else
        tmp = x_m * ((y / z) - t)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -9.0) || !(z <= 0.0005)) {
		tmp = x_m * ((y + t) / z);
	} else {
		tmp = x_m * ((y / z) - t);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -9.0) or not (z <= 0.0005):
		tmp = x_m * ((y + t) / z)
	else:
		tmp = x_m * ((y / z) - t)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -9.0) || !(z <= 0.0005))
		tmp = Float64(x_m * Float64(Float64(y + t) / z));
	else
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -9.0) || ~((z <= 0.0005)))
		tmp = x_m * ((y + t) / z);
	else
		tmp = x_m * ((y / z) - t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -9.0], N[Not[LessEqual[z, 0.0005]], $MachinePrecision]], N[(x$95$m * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -9 or 5.0000000000000001e-4 < z

   1. Initial program 98.4%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 98.4%

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

      2. associate-/r/ 98.3%

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in z around inf 97.6%

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

      1. sub-neg 97.6%

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

      2. mul-1-neg 97.6%

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

      3. remove-double-neg 97.6%

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

   7. Simplified 97.6%

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

   if -9 < z < 5.0000000000000001e-4

   1. Initial program 93.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 92.8%

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

      1. +-commutative 92.8%

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

      2. associate-*r/ 89.4%

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

      3. *-commutative 89.4%

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

      4. associate-*r* 89.4%

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

      5. neg-mul-1 89.4%

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

      6. distribute-rgt-out 93.1%

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

      7. unsub-neg 93.1%

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

   5. Simplified 93.1%

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

4. Final simplification 95.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \lor \neg \left(z \leq 0.0005\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 93.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -9:\\ \;\;\;\;\frac{x\_m}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 0.0005:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y + t}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= z -9.0)
    (/ x_m (/ z (+ y t)))
    (if (<= z 0.0005) (* x_m (- (/ y z) t)) (* x_m (/ (+ y t) z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9.0) {
		tmp = x_m / (z / (y + t));
	} else if (z <= 0.0005) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = x_m * ((y + t) / z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-9.0d0)) then
        tmp = x_m / (z / (y + t))
    else if (z <= 0.0005d0) then
        tmp = x_m * ((y / z) - t)
    else
        tmp = x_m * ((y + t) / z)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (z <= -9.0) {
		tmp = x_m / (z / (y + t));
	} else if (z <= 0.0005) {
		tmp = x_m * ((y / z) - t);
	} else {
		tmp = x_m * ((y + t) / z);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if z <= -9.0:
		tmp = x_m / (z / (y + t))
	elif z <= 0.0005:
		tmp = x_m * ((y / z) - t)
	else:
		tmp = x_m * ((y + t) / z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (z <= -9.0)
		tmp = Float64(x_m / Float64(z / Float64(y + t)));
	elseif (z <= 0.0005)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	else
		tmp = Float64(x_m * Float64(Float64(y + t) / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (z <= -9.0)
		tmp = x_m / (z / (y + t));
	elseif (z <= 0.0005)
		tmp = x_m * ((y / z) - t);
	else
		tmp = x_m * ((y + t) / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[z, -9.0], N[(x$95$m / N[(z / N[(y + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.0005], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if z < -9

   1. Initial program 97.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf 88.1%

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

      1. associate-/l* 96.6%

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

      2. cancel-sign-sub-inv 96.6%

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

      3. metadata-eval 96.6%

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

      4. *-lft-identity 96.6%

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

      5. +-commutative 96.6%

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

   5. Simplified 96.6%

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

   if -9 < z < 5.0000000000000001e-4

   1. Initial program 93.6%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 92.8%

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

      1. +-commutative 92.8%

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

      2. associate-*r/ 89.4%

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

      3. *-commutative 89.4%

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

      4. associate-*r* 89.4%

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

      5. neg-mul-1 89.4%

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

      6. distribute-rgt-out 93.1%

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

      7. unsub-neg 93.1%

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

   5. Simplified 93.1%

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

   if 5.0000000000000001e-4 < z

   1. Initial program 98.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 98.8%

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

      2. associate-/r/ 98.8%

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

   4. Applied egg-rr 98.8%

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

   5. Taylor expanded in z around inf 98.5%

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

      1. sub-neg 98.5%

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

      2. mul-1-neg 98.5%

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

      3. remove-double-neg 98.5%

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

   7. Simplified 98.5%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9:\\ \;\;\;\;\frac{x}{\frac{z}{y + t}}\\ \mathbf{elif}\;z \leq 0.0005:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -0.022 \lor \neg \left(z \leq 22500\right):\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -0.022) (not (<= z 22500.0))) (* x_m (/ t z)) (* x_m (- t)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -0.022) || !(z <= 22500.0)) {
		tmp = x_m * (t / z);
	} else {
		tmp = x_m * -t;
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-0.022d0)) .or. (.not. (z <= 22500.0d0))) then
        tmp = x_m * (t / z)
    else
        tmp = x_m * -t
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((z <= -0.022) || !(z <= 22500.0)) {
		tmp = x_m * (t / z);
	} else {
		tmp = x_m * -t;
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (z <= -0.022) or not (z <= 22500.0):
		tmp = x_m * (t / z)
	else:
		tmp = x_m * -t
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((z <= -0.022) || !(z <= 22500.0))
		tmp = Float64(x_m * Float64(t / z));
	else
		tmp = Float64(x_m * Float64(-t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((z <= -0.022) || ~((z <= 22500.0)))
		tmp = x_m * (t / z);
	else
		tmp = x_m * -t;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -0.022], N[Not[LessEqual[z, 22500.0]], $MachinePrecision]], N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * (-t)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -0.021999999999999999 or 22500 < z

   1. Initial program 98.3%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 98.3%

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

      2. associate-/r/ 98.3%

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

   4. Applied egg-rr 98.3%

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

   5. Taylor expanded in z around inf 86.9%

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

      1. associate-/l* 97.5%

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

      2. sub-neg 97.5%

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

      3. mul-1-neg 97.5%

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

      4. remove-double-neg 97.5%

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

   7. Simplified 97.5%

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

   8. Taylor expanded in y around 0 51.8%

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

      1. *-commutative 51.8%

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

      2. associate-*r/ 56.1%

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

   10. Simplified 56.1%

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

   if -0.021999999999999999 < z < 22500

   1. Initial program 93.7%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 93.0%

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

      1. +-commutative 93.0%

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

      2. associate-*r/ 89.7%

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

      3. *-commutative 89.7%

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

      4. associate-*r* 89.7%

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

      5. neg-mul-1 89.7%

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

      6. distribute-rgt-out 93.3%

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

      7. unsub-neg 93.3%

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

   5. Simplified 93.3%

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

   6. Taylor expanded in y around 0 31.7%

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

      1. associate-*r* 31.7%

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

      2. mul-1-neg 31.7%

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

   8. Simplified 31.7%

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

4. Final simplification 42.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.022 \lor \neg \left(z \leq 22500\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 64.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-129} \lor \neg \left(y \leq 3.3 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -1.5e-129) (not (<= y 3.3e-47)))
    (* (/ x_m z) y)
    (* x_m (/ t z)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -1.5e-129) || !(y <= 3.3e-47)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m * (t / z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1.5d-129)) .or. (.not. (y <= 3.3d-47))) then
        tmp = (x_m / z) * y
    else
        tmp = x_m * (t / z)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -1.5e-129) || !(y <= 3.3e-47)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m * (t / z);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -1.5e-129) or not (y <= 3.3e-47):
		tmp = (x_m / z) * y
	else:
		tmp = x_m * (t / z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -1.5e-129) || !(y <= 3.3e-47))
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = Float64(x_m * Float64(t / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -1.5e-129) || ~((y <= 3.3e-47)))
		tmp = (x_m / z) * y;
	else
		tmp = x_m * (t / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -1.5e-129], N[Not[LessEqual[y, 3.3e-47]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.4999999999999999e-129 or 3.30000000000000004e-47 < y

   1. Initial program 94.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 76.0%

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

      1. associate-/l* 76.1%

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

      2. associate-/r/ 78.8%

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

   5. Simplified 78.8%

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

   if -1.4999999999999999e-129 < y < 3.30000000000000004e-47

   1. Initial program 97.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 97.8%

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

      2. associate-/r/ 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in z around inf 63.2%

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

      1. associate-/l* 63.5%

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

      2. sub-neg 63.5%

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

      3. mul-1-neg 63.5%

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

      4. remove-double-neg 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in y around 0 54.1%

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

      1. *-commutative 54.1%

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

      2. associate-*r/ 55.5%

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

   10. Simplified 55.5%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.5 \cdot 10^{-129} \lor \neg \left(y \leq 3.3 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-129} \lor \neg \left(y \leq 3.2 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x\_m}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m}{\frac{z}{t}}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -1.25e-129) (not (<= y 3.2e-47)))
    (* (/ x_m z) y)
    (/ x_m (/ z t)))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -1.25e-129) || !(y <= 3.2e-47)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m / (z / t);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1.25d-129)) .or. (.not. (y <= 3.2d-47))) then
        tmp = (x_m / z) * y
    else
        tmp = x_m / (z / t)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((y <= -1.25e-129) || !(y <= 3.2e-47)) {
		tmp = (x_m / z) * y;
	} else {
		tmp = x_m / (z / t);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (y <= -1.25e-129) or not (y <= 3.2e-47):
		tmp = (x_m / z) * y
	else:
		tmp = x_m / (z / t)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((y <= -1.25e-129) || !(y <= 3.2e-47))
		tmp = Float64(Float64(x_m / z) * y);
	else
		tmp = Float64(x_m / Float64(z / t));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((y <= -1.25e-129) || ~((y <= 3.2e-47)))
		tmp = (x_m / z) * y;
	else
		tmp = x_m / (z / t);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[y, -1.25e-129], N[Not[LessEqual[y, 3.2e-47]], $MachinePrecision]], N[(N[(x$95$m / z), $MachinePrecision] * y), $MachinePrecision], N[(x$95$m / N[(z / t), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -1.25000000000000007e-129 or 3.1999999999999999e-47 < y

   1. Initial program 94.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 76.0%

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

      1. associate-/l* 76.1%

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

      2. associate-/r/ 78.8%

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

   5. Simplified 78.8%

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

   if -1.25000000000000007e-129 < y < 3.1999999999999999e-47

   1. Initial program 97.9%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 97.8%

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

      2. associate-/r/ 97.8%

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

   4. Applied egg-rr 97.8%

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

   5. Taylor expanded in z around inf 63.2%

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

      1. associate-/l* 63.5%

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

      2. sub-neg 63.5%

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

      3. mul-1-neg 63.5%

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

      4. remove-double-neg 63.5%

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

   7. Simplified 63.5%

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

   8. Taylor expanded in y around 0 55.6%

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

4. Final simplification 70.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.25 \cdot 10^{-129} \lor \neg \left(y \leq 3.2 \cdot 10^{-47}\right):\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 66.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x_m = \left|x\right| \\ x_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{+193}:\\ \;\;\;\;x\_m \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 10^{+79}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

x_m = (fabs.f64 x)
x_s = (copysign.f64 1 x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -1.9e+193)
    (* x_m (- t))
    (if (<= t 1e+79) (* x_m (/ y z)) (* x_m (/ t z))))))

C

x_m = fabs(x);
x_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -1.9e+193) {
		tmp = x_m * -t;
	} else if (t <= 1e+79) {
		tmp = x_m * (y / z);
	} else {
		tmp = x_m * (t / z);
	}
	return x_s * tmp;
}

Fortran

x_m = abs(x)
x_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.9d+193)) then
        tmp = x_m * -t
    else if (t <= 1d+79) then
        tmp = x_m * (y / z)
    else
        tmp = x_m * (t / z)
    end if
    code = x_s * tmp
end function

Java

x_m = Math.abs(x);
x_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (t <= -1.9e+193) {
		tmp = x_m * -t;
	} else if (t <= 1e+79) {
		tmp = x_m * (y / z);
	} else {
		tmp = x_m * (t / z);
	}
	return x_s * tmp;
}

Python

x_m = math.fabs(x)
x_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if t <= -1.9e+193:
		tmp = x_m * -t
	elif t <= 1e+79:
		tmp = x_m * (y / z)
	else:
		tmp = x_m * (t / z)
	return x_s * tmp

Julia

x_m = abs(x)
x_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (t <= -1.9e+193)
		tmp = Float64(x_m * Float64(-t));
	elseif (t <= 1e+79)
		tmp = Float64(x_m * Float64(y / z));
	else
		tmp = Float64(x_m * Float64(t / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (t <= -1.9e+193)
		tmp = x_m * -t;
	elseif (t <= 1e+79)
		tmp = x_m * (y / z);
	else
		tmp = x_m * (t / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x_m = N[Abs[x], $MachinePrecision]
x_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[t, -1.9e+193], N[(x$95$m * (-t)), $MachinePrecision], If[LessEqual[t, 1e+79], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -1.89999999999999986e193

   1. Initial program 99.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0 60.3%

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

      1. +-commutative 60.3%

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

      2. associate-*r/ 60.4%

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

      3. *-commutative 60.4%

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

      4. associate-*r* 60.4%

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

      5. neg-mul-1 60.4%

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

      6. distribute-rgt-out 63.9%

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

      7. unsub-neg 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in y around 0 53.5%

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

      1. associate-*r* 53.5%

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

      2. mul-1-neg 53.5%

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

   8. Simplified 53.5%

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

   if -1.89999999999999986e193 < t < 9.99999999999999967e78

   1. Initial program 95.0%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf 77.0%

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

      1. associate-*r/ 77.5%

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

   5. Simplified 77.5%

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

   if 9.99999999999999967e78 < t

   1. Initial program 97.0%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. clear-num 97.0%

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

      2. associate-/r/ 97.0%

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

   4. Applied egg-rr 97.0%

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

   5. Taylor expanded in z around inf 44.0%

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

      1. associate-/l* 59.4%

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

      2. sub-neg 59.4%

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

      3. mul-1-neg 59.4%

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

      4. remove-double-neg 59.4%

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

   7. Simplified 59.4%

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

   8. Taylor expanded in y around 0 31.2%

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

      1. *-commutative 31.2%

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

      2. associate-*r/ 44.0%

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

   10. Simplified 44.0%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.9 \cdot 10^{+193}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 10^{+79}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 23.4% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

x_m = abs(x);
x_s = sign(x) * abs(1.0);
function tmp = code(x_s, x_m, y, z, t)
	tmp = x_s * (x_m * -t);
end

Wolfram

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

Derivation

1. Initial program 95.8%

   \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
2. Add Preprocessing

3. Taylor expanded in z around 0 66.2%

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

   1. +-commutative 66.2%

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

   2. associate-*r/ 65.6%

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

   3. *-commutative 65.6%

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

   4. associate-*r* 65.6%

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

   5. neg-mul-1 65.6%

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

   6. distribute-rgt-out 67.9%

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

   7. unsub-neg 67.9%

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

5. Simplified 67.9%

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

6. Taylor expanded in y around 0 22.3%

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

   1. associate-*r* 22.3%

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

   2. mul-1-neg 22.3%

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

8. Simplified 22.3%

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

9. Final simplification 22.3%

   \[\leadsto x \cdot \left(-t\right) \]
10. Add Preprocessing

Developer target: 95.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ t_2 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t\_2 < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))))
        (t_2 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (< t_2 -7.623226303312042e-196)
     t_1
     (if (< t_2 1.4133944927702302e-211)
       (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z))))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * ((y / z) - (t * (1.0d0 / (1.0d0 - z))))
    t_2 = x * ((y / z) - (t / (1.0d0 - z)))
    if (t_2 < (-7.623226303312042d-196)) then
        tmp = t_1
    else if (t_2 < 1.4133944927702302d-211) then
        tmp = ((y * x) / z) + -((t * x) / (1.0d0 - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))))
	t_2 = x * ((y / z) - (t / (1.0 - z)))
	tmp = 0
	if t_2 < -7.623226303312042e-196:
		tmp = t_1
	elif t_2 < 1.4133944927702302e-211:
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y / z) - Float64(t * Float64(1.0 / Float64(1.0 - z)))))
	t_2 = Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = Float64(Float64(Float64(y * x) / z) + Float64(-Float64(Float64(t * x) / Float64(1.0 - z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	t_2 = x * ((y / z) - (t / (1.0 - z)));
	tmp = 0.0;
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -7.623226303312042e-196], t$95$1, If[Less[t$95$2, 1.4133944927702302e-211], N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] + (-N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024031 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))