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

Percentage Accurate: 94.8% → 96.8%

Time: 11.3s

Alternatives: 13

Speedup: 0.6×

• 21.8% 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: 96.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;x\_m \leq 2000000000000:\\ \;\;\;\;\frac{x\_m \cdot y}{z} - \frac{x\_m \cdot t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} + \frac{t}{-1 + z}\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= x_m 2000000000000.0)
    (- (/ (* x_m y) z) (/ (* x_m t) (- 1.0 z)))
    (* x_m (+ (/ y z) (/ t (+ -1.0 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 (x_m <= 2000000000000.0) {
		tmp = ((x_m * y) / z) - ((x_m * t) / (1.0 - z));
	} else {
		tmp = x_m * ((y / z) + (t / (-1.0 + 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 (x_m <= 2000000000000.0d0) then
        tmp = ((x_m * y) / z) - ((x_m * t) / (1.0d0 - z))
    else
        tmp = x_m * ((y / z) + (t / ((-1.0d0) + 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 (x_m <= 2000000000000.0) {
		tmp = ((x_m * y) / z) - ((x_m * t) / (1.0 - z));
	} else {
		tmp = x_m * ((y / z) + (t / (-1.0 + 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 x_m <= 2000000000000.0:
		tmp = ((x_m * y) / z) - ((x_m * t) / (1.0 - z))
	else:
		tmp = x_m * ((y / z) + (t / (-1.0 + 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 (x_m <= 2000000000000.0)
		tmp = Float64(Float64(Float64(x_m * y) / z) - Float64(Float64(x_m * t) / Float64(1.0 - z)));
	else
		tmp = Float64(x_m * Float64(Float64(y / z) + Float64(t / Float64(-1.0 + 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 (x_m <= 2000000000000.0)
		tmp = ((x_m * y) / z) - ((x_m * t) / (1.0 - z));
	else
		tmp = x_m * ((y / z) + (t / (-1.0 + 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[x$95$m, 2000000000000.0], N[(N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision] - N[(N[(x$95$m * t), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(N[(y / z), $MachinePrecision] + N[(t / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if x < 2e12

   1. Initial program 95.4%

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

   3. Taylor expanded in y around 0 92.2%

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

   if 2e12 < x

   1. Initial program 96.9%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 2000000000000:\\ \;\;\;\;\frac{x \cdot y}{z} - \frac{x \cdot t}{1 - z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{-1 + z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.6% accurate, 0.5× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 (- INFINITY)) (* y (/ x_m z)) (* x_m 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 = (y / z) + (t / (-1.0 + z));
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (x_m / z);
	} else {
		tmp = x_m * t_1;
	}
	return x_s * tmp;
}

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 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x_m / z);
	} else {
		tmp = x_m * 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 = (y / z) + (t / (-1.0 + z))
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * (x_m / z)
	else:
		tmp = x_m * 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(Float64(y / z) + Float64(t / Float64(-1.0 + z)))
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(x_m / z));
	else
		tmp = Float64(x_m * 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 = (y / z) + (t / (-1.0 + z));
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * (x_m / z);
	else
		tmp = x_m * 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[(N[(y / z), $MachinePrecision] + N[(t / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * t$95$1), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0

   1. Initial program 81.5%

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

   3. Taylor expanded in y around inf 95.9%

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

      1. associate-*r/ 95.9%

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

      2. associate-*r* 95.9%

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

      3. neg-mul-1 95.9%

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

      4. *-commutative 95.9%

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

      5. times-frac 99.9%

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

      6. distribute-frac-neg 99.9%

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

      7. distribute-neg-frac2 99.9%

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

      8. neg-sub0 99.9%

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

      9. associate--r- 99.9%

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

      10. metadata-eval 99.9%

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

   5. Simplified 99.9%

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

   6. Taylor expanded in t around 0 99.9%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 97.3%

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

4. Final simplification 97.6%

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

Alternative 3: 73.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ \begin{array}{l} t_1 := x\_m \cdot \frac{y}{z}\\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -1.85 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 16500:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+131}:\\ \;\;\;\;t \cdot \frac{x\_m}{-1 + z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (let* ((t_1 (* x_m (/ y z))))
   (*
    x_s
    (if (<= z -1.85e+92)
      t_1
      (if (<= z 16500.0)
        (* x_m (- (/ y z) t))
        (if (<= z 1.95e+131) (* t (/ x_m (+ -1.0 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 = x_m * (y / z);
	double tmp;
	if (z <= -1.85e+92) {
		tmp = t_1;
	} else if (z <= 16500.0) {
		tmp = x_m * ((y / z) - t);
	} else if (z <= 1.95e+131) {
		tmp = t * (x_m / (-1.0 + z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x_m * (y / z)
    if (z <= (-1.85d+92)) then
        tmp = t_1
    else if (z <= 16500.0d0) then
        tmp = x_m * ((y / z) - t)
    else if (z <= 1.95d+131) then
        tmp = t * (x_m / ((-1.0d0) + z))
    else
        tmp = t_1
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double t_1 = x_m * (y / z);
	double tmp;
	if (z <= -1.85e+92) {
		tmp = t_1;
	} else if (z <= 16500.0) {
		tmp = x_m * ((y / z) - t);
	} else if (z <= 1.95e+131) {
		tmp = t * (x_m / (-1.0 + z));
	} else {
		tmp = t_1;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	t_1 = x_m * (y / z)
	tmp = 0
	if z <= -1.85e+92:
		tmp = t_1
	elif z <= 16500.0:
		tmp = x_m * ((y / z) - t)
	elif z <= 1.95e+131:
		tmp = t * (x_m / (-1.0 + z))
	else:
		tmp = t_1
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	t_1 = Float64(x_m * Float64(y / z))
	tmp = 0.0
	if (z <= -1.85e+92)
		tmp = t_1;
	elseif (z <= 16500.0)
		tmp = Float64(x_m * Float64(Float64(y / z) - t));
	elseif (z <= 1.95e+131)
		tmp = Float64(t * Float64(x_m / Float64(-1.0 + z)));
	else
		tmp = t_1;
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	t_1 = x_m * (y / z);
	tmp = 0.0;
	if (z <= -1.85e+92)
		tmp = t_1;
	elseif (z <= 16500.0)
		tmp = x_m * ((y / z) - t);
	elseif (z <= 1.95e+131)
		tmp = t * (x_m / (-1.0 + z));
	else
		tmp = t_1;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := Block[{t$95$1 = N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]}, N[(x$95$s * If[LessEqual[z, -1.85e+92], t$95$1, If[LessEqual[z, 16500.0], N[(x$95$m * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.95e+131], N[(t * N[(x$95$m / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if z < -1.84999999999999999e92 or 1.95e131 < z

   1. Initial program 98.3%

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

   3. Taylor expanded in y around inf 64.2%

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

      1. associate-*r/ 70.3%

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

   5. Simplified 70.3%

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

   if -1.84999999999999999e92 < z < 16500

   1. Initial program 94.6%

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

   3. Taylor expanded in z around 0 91.7%

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

   if 16500 < z < 1.95e131

   1. Initial program 96.0%

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

   3. Taylor expanded in y around 0 67.5%

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

      1. mul-1-neg 67.5%

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

      2. associate-/l* 74.5%

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

      3. distribute-rgt-neg-in 74.5%

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

      4. distribute-neg-frac2 74.5%

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

      5. neg-sub0 74.5%

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

      6. associate--r- 74.5%

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

      7. metadata-eval 74.5%

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

   5. Simplified 74.5%

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

Alternative 4: 95.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -118000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x\_m \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot \left(y - t \cdot z\right)}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -118000.0) (not (<= z 1.0)))
    (* x_m (/ (+ t y) z))
    (/ (* x_m (- y (* t z))) 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 <= -118000.0) || !(z <= 1.0)) {
		tmp = x_m * ((t + y) / z);
	} else {
		tmp = (x_m * (y - (t * z))) / 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 <= (-118000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x_m * ((t + y) / z)
    else
        tmp = (x_m * (y - (t * z))) / 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 <= -118000.0) || !(z <= 1.0)) {
		tmp = x_m * ((t + y) / z);
	} else {
		tmp = (x_m * (y - (t * z))) / 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 <= -118000.0) or not (z <= 1.0):
		tmp = x_m * ((t + y) / z)
	else:
		tmp = (x_m * (y - (t * z))) / 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 <= -118000.0) || !(z <= 1.0))
		tmp = Float64(x_m * Float64(Float64(t + y) / z));
	else
		tmp = Float64(Float64(x_m * Float64(y - Float64(t * z))) / 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 <= -118000.0) || ~((z <= 1.0)))
		tmp = x_m * ((t + y) / z);
	else
		tmp = (x_m * (y - (t * z))) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[z, -118000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x$95$m * N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < -118000 or 1 < z

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf 84.8%

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

      1. associate-/l* 97.2%

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

      2. neg-mul-1 97.2%

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

      3. sub-neg 97.2%

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

      4. remove-double-neg 97.2%

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

      5. neg-mul-1 97.2%

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

      6. neg-mul-1 97.2%

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

      7. distribute-lft-in 97.2%

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

      8. neg-mul-1 97.2%

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

      9. sub-neg 97.2%

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

      10. *-commutative 97.2%

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

      11. associate-*l/ 97.2%

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

      12. *-commutative 97.2%

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

      13. associate-*r/ 97.2%

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

      14. sub-neg 97.2%

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

      15. neg-mul-1 97.2%

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

      16. distribute-lft-in 97.2%

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

      17. neg-mul-1 97.2%

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

      18. remove-double-neg 97.2%

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

      19. neg-mul-1 97.2%

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

      20. remove-double-neg 97.2%

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

      21. +-commutative 97.2%

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

   5. Simplified 97.2%

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

   if -118000 < z < 1

   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 94.1%

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

      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 z around 0 90.7%

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

      1. +-commutative 90.7%

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

      2. *-commutative 90.7%

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

      3. mul-1-neg 90.7%

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

      4. associate-*r* 94.2%

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

      5. *-commutative 94.2%

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

      6. distribute-lft-neg-out 94.2%

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

      7. associate-*r* 94.2%

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

      8. distribute-rgt-out 96.3%

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

      9. distribute-lft-neg-in 96.3%

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

      10. unsub-neg 96.3%

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

   7. Simplified 96.3%

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

4. Final simplification 96.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -118000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y - t \cdot z\right)}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -118000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x\_m \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -118000.0) (not (<= z 1.0)))
    (* x_m (/ (+ t y) 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 <= -118000.0) || !(z <= 1.0)) {
		tmp = x_m * ((t + y) / 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 <= (-118000.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x_m * ((t + y) / 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 <= -118000.0) || !(z <= 1.0)) {
		tmp = x_m * ((t + y) / 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 <= -118000.0) or not (z <= 1.0):
		tmp = x_m * ((t + y) / 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 <= -118000.0) || !(z <= 1.0))
		tmp = Float64(x_m * Float64(Float64(t + y) / 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 <= -118000.0) || ~((z <= 1.0)))
		tmp = x_m * ((t + y) / 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, -118000.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x$95$m * N[(N[(t + y), $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 < -118000 or 1 < z

   1. Initial program 97.9%

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

   3. Taylor expanded in z around inf 84.8%

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

      1. associate-/l* 97.2%

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

      2. neg-mul-1 97.2%

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

      3. sub-neg 97.2%

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

      4. remove-double-neg 97.2%

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

      5. neg-mul-1 97.2%

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

      6. neg-mul-1 97.2%

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

      7. distribute-lft-in 97.2%

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

      8. neg-mul-1 97.2%

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

      9. sub-neg 97.2%

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

      10. *-commutative 97.2%

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

      11. associate-*l/ 97.2%

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

      12. *-commutative 97.2%

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

      13. associate-*r/ 97.2%

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

      14. sub-neg 97.2%

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

      15. neg-mul-1 97.2%

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

      16. distribute-lft-in 97.2%

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

      17. neg-mul-1 97.2%

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

      18. remove-double-neg 97.2%

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

      19. neg-mul-1 97.2%

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

      20. remove-double-neg 97.2%

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

      21. +-commutative 97.2%

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

   5. Simplified 97.2%

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

   if -118000 < z < 1

   1. Initial program 94.2%

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

   3. Taylor expanded in z around 0 93.7%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -118000 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{t + y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -2.55 \cdot 10^{+67} \lor \neg \left(t \leq 9.5 \cdot 10^{+43}\right):\\ \;\;\;\;t \cdot \frac{x\_m}{-1 + z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= t -2.55e+67) (not (<= t 9.5e+43)))
    (* t (/ x_m (+ -1.0 z)))
    (/ (* x_m y) z))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t <= -2.55e+67) || !(t <= 9.5e+43)) {
		tmp = t * (x_m / (-1.0 + z));
	} else {
		tmp = (x_m * y) / z;
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.55d+67)) .or. (.not. (t <= 9.5d+43))) then
        tmp = t * (x_m / ((-1.0d0) + z))
    else
        tmp = (x_m * y) / z
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if ((t <= -2.55e+67) || !(t <= 9.5e+43)) {
		tmp = t * (x_m / (-1.0 + z));
	} else {
		tmp = (x_m * y) / z;
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if (t <= -2.55e+67) or not (t <= 9.5e+43):
		tmp = t * (x_m / (-1.0 + z))
	else:
		tmp = (x_m * y) / z
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if ((t <= -2.55e+67) || !(t <= 9.5e+43))
		tmp = Float64(t * Float64(x_m / Float64(-1.0 + z)));
	else
		tmp = Float64(Float64(x_m * y) / z);
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if ((t <= -2.55e+67) || ~((t <= 9.5e+43)))
		tmp = t * (x_m / (-1.0 + z));
	else
		tmp = (x_m * y) / z;
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[Or[LessEqual[t, -2.55e+67], N[Not[LessEqual[t, 9.5e+43]], $MachinePrecision]], N[(t * N[(x$95$m / N[(-1.0 + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < -2.5500000000000001e67 or 9.5000000000000004e43 < t

   1. Initial program 97.3%

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

   3. Taylor expanded in y around 0 65.4%

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

      1. mul-1-neg 65.4%

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

      2. associate-/l* 72.1%

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

      3. distribute-rgt-neg-in 72.1%

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

      4. distribute-neg-frac2 72.1%

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

      5. neg-sub0 72.1%

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

      6. associate--r- 72.1%

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

      7. metadata-eval 72.1%

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

   5. Simplified 72.1%

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

   if -2.5500000000000001e67 < t < 9.5000000000000004e43

   1. Initial program 94.6%

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

   3. Taylor expanded in y around inf 83.4%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.55 \cdot 10^{+67} \lor \neg \left(t \leq 9.5 \cdot 10^{+43}\right):\\ \;\;\;\;t \cdot \frac{x}{-1 + z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 64.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-142} \lor \neg \left(y \leq 1.95 \cdot 10^{-142}\right):\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= y -6.2e-142) (not (<= y 1.95e-142)))
    (* 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 ((y <= -6.2e-142) || !(y <= 1.95e-142)) {
		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 ((y <= (-6.2d-142)) .or. (.not. (y <= 1.95d-142))) 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 ((y <= -6.2e-142) || !(y <= 1.95e-142)) {
		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 (y <= -6.2e-142) or not (y <= 1.95e-142):
		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 ((y <= -6.2e-142) || !(y <= 1.95e-142))
		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 ((y <= -6.2e-142) || ~((y <= 1.95e-142)))
		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[Or[LessEqual[y, -6.2e-142], N[Not[LessEqual[y, 1.95e-142]], $MachinePrecision]], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if y < -6.2e-142 or 1.9500000000000002e-142 < y

   1. Initial program 95.1%

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

   3. Taylor expanded in y around inf 72.1%

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

      1. associate-*r/ 71.3%

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

   5. Simplified 71.3%

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

   if -6.2e-142 < y < 1.9500000000000002e-142

   1. Initial program 98.2%

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

      1. clear-num 98.0%

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

      2. associate-/r/ 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in z around inf 55.9%

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

      1. associate-/l* 60.5%

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

      2. neg-mul-1 60.5%

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

      3. sub-neg 60.5%

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

      4. remove-double-neg 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in y around 0 58.9%

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

4. Final simplification 68.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.2 \cdot 10^{-142} \lor \neg \left(y \leq 1.95 \cdot 10^{-142}\right):\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 43.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-65} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z -3.1e-65) (not (<= z 1.0))) (* 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 <= -3.1e-65) || !(z <= 1.0)) {
		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 <= (-3.1d-65)) .or. (.not. (z <= 1.0d0))) 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 <= -3.1e-65) || !(z <= 1.0)) {
		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 <= -3.1e-65) or not (z <= 1.0):
		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 <= -3.1e-65) || !(z <= 1.0))
		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 <= -3.1e-65) || ~((z <= 1.0)))
		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, -3.1e-65], N[Not[LessEqual[z, 1.0]], $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 < -3.10000000000000016e-65 or 1 < z

   1. Initial program 98.1%

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

      1. clear-num 98.0%

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

      2. associate-/r/ 98.0%

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

   4. Applied egg-rr 98.0%

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

   5. Taylor expanded in z around inf 85.2%

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

      1. associate-/l* 96.2%

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

      2. neg-mul-1 96.2%

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

      3. sub-neg 96.2%

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

      4. remove-double-neg 96.2%

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

   7. Simplified 96.2%

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

   8. Taylor expanded in y around 0 53.3%

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

   if -3.10000000000000016e-65 < z < 1

   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 93.1%

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

   4. Taylor expanded in y around 0 39.0%

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

      1. neg-mul-1 39.0%

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

      2. *-commutative 39.0%

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

      3. distribute-rgt-neg-out 39.0%

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

   6. Simplified 39.0%

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

4. Final simplification 46.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-65} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 39.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-306} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;t \cdot \frac{x\_m}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (or (<= z 2.7e-306) (not (<= z 1.0))) (* t (/ x_m 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 <= 2.7e-306) || !(z <= 1.0)) {
		tmp = t * (x_m / 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 <= 2.7d-306) .or. (.not. (z <= 1.0d0))) then
        tmp = t * (x_m / 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 <= 2.7e-306) || !(z <= 1.0)) {
		tmp = t * (x_m / 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 <= 2.7e-306) or not (z <= 1.0):
		tmp = t * (x_m / 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 <= 2.7e-306) || !(z <= 1.0))
		tmp = Float64(t * Float64(x_m / 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 <= 2.7e-306) || ~((z <= 1.0)))
		tmp = t * (x_m / 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, 2.7e-306], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(t * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * (-t)), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 2.70000000000000009e-306 or 1 < z

   1. Initial program 97.6%

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

      1. clear-num 97.5%

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

      2. associate-/r/ 97.5%

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

   4. Applied egg-rr 97.5%

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

   5. Taylor expanded in z around inf 80.2%

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

      1. associate-/l* 87.0%

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

      2. neg-mul-1 87.0%

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

      3. sub-neg 87.0%

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

      4. remove-double-neg 87.0%

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

   7. Simplified 87.0%

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

   8. Taylor expanded in y around 0 40.8%

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

      1. associate-/l* 47.2%

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

   10. Simplified 47.2%

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

   if 2.70000000000000009e-306 < z < 1

   1. Initial program 91.9%

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

   3. Taylor expanded in z around 0 91.0%

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

   4. Taylor expanded in y around 0 41.2%

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

      1. neg-mul-1 41.2%

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

      2. *-commutative 41.2%

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

      3. distribute-rgt-neg-out 41.2%

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

   6. Simplified 41.2%

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

4. Final simplification 45.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 2.7 \cdot 10^{-306} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;t \leq -5.9 \cdot 10^{+186}:\\ \;\;\;\;x\_m \cdot \left(-t\right)\\ \mathbf{elif}\;t \leq 2.15 \cdot 10^{+71}:\\ \;\;\;\;\frac{x\_m \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= t -5.9e+186)
    (* x_m (- t))
    (if (<= t 2.15e+71) (/ (* 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 <= -5.9e+186) {
		tmp = x_m * -t;
	} else if (t <= 2.15e+71) {
		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 <= (-5.9d+186)) then
        tmp = x_m * -t
    else if (t <= 2.15d+71) 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 <= -5.9e+186) {
		tmp = x_m * -t;
	} else if (t <= 2.15e+71) {
		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 <= -5.9e+186:
		tmp = x_m * -t
	elif t <= 2.15e+71:
		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 <= -5.9e+186)
		tmp = Float64(x_m * Float64(-t));
	elseif (t <= 2.15e+71)
		tmp = Float64(Float64(x_m * 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 <= -5.9e+186)
		tmp = x_m * -t;
	elseif (t <= 2.15e+71)
		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, -5.9e+186], N[(x$95$m * (-t)), $MachinePrecision], If[LessEqual[t, 2.15e+71], N[(N[(x$95$m * y), $MachinePrecision] / z), $MachinePrecision], N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < -5.89999999999999982e186

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0 71.6%

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

   4. Taylor expanded in y around 0 55.6%

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

      1. neg-mul-1 55.6%

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

      2. *-commutative 55.6%

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

      3. distribute-rgt-neg-out 55.6%

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

   6. Simplified 55.6%

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

   if -5.89999999999999982e186 < t < 2.14999999999999992e71

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf 79.9%

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

   if 2.14999999999999992e71 < t

   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 98.2%

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

      2. associate-/r/ 98.4%

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

   4. Applied egg-rr 98.4%

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

   5. Taylor expanded in z around inf 51.4%

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

      1. associate-/l* 58.4%

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

      2. neg-mul-1 58.4%

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

      3. sub-neg 58.4%

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

      4. remove-double-neg 58.4%

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

   7. Simplified 58.4%

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

   8. Taylor expanded in y around 0 48.3%

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

Alternative 11: 64.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} x\_m = \left|x\right| \\ x\_s = \mathsf{copysign}\left(1, x\right) \\ x\_s \cdot \begin{array}{l} \mathbf{if}\;y \leq -4.5 \cdot 10^{-141}:\\ \;\;\;\;y \cdot \frac{x\_m}{z}\\ \mathbf{elif}\;y \leq 6 \cdot 10^{-142}:\\ \;\;\;\;x\_m \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x\_m \cdot \frac{y}{z}\\ \end{array} \end{array} \]

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (*
  x_s
  (if (<= y -4.5e-141)
    (* y (/ x_m z))
    (if (<= y 6e-142) (* x_m (/ t z)) (* x_m (/ y z))))))

C

x\_m = fabs(x);
x\_s = copysign(1.0, x);
double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -4.5e-141) {
		tmp = y * (x_m / z);
	} else if (y <= 6e-142) {
		tmp = x_m * (t / z);
	} else {
		tmp = x_m * (y / z);
	}
	return x_s * tmp;
}

Fortran

x\_m = abs(x)
x\_s = copysign(1.0d0, x)
real(8) function code(x_s, x_m, y, z, t)
    real(8), intent (in) :: x_s
    real(8), intent (in) :: x_m
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-4.5d-141)) then
        tmp = y * (x_m / z)
    else if (y <= 6d-142) then
        tmp = x_m * (t / z)
    else
        tmp = x_m * (y / z)
    end if
    code = x_s * tmp
end function

Java

x\_m = Math.abs(x);
x\_s = Math.copySign(1.0, x);
public static double code(double x_s, double x_m, double y, double z, double t) {
	double tmp;
	if (y <= -4.5e-141) {
		tmp = y * (x_m / z);
	} else if (y <= 6e-142) {
		tmp = x_m * (t / z);
	} else {
		tmp = x_m * (y / z);
	}
	return x_s * tmp;
}

Python

x\_m = math.fabs(x)
x\_s = math.copysign(1.0, x)
def code(x_s, x_m, y, z, t):
	tmp = 0
	if y <= -4.5e-141:
		tmp = y * (x_m / z)
	elif y <= 6e-142:
		tmp = x_m * (t / z)
	else:
		tmp = x_m * (y / z)
	return x_s * tmp

Julia

x\_m = abs(x)
x\_s = copysign(1.0, x)
function code(x_s, x_m, y, z, t)
	tmp = 0.0
	if (y <= -4.5e-141)
		tmp = Float64(y * Float64(x_m / z));
	elseif (y <= 6e-142)
		tmp = Float64(x_m * Float64(t / z));
	else
		tmp = Float64(x_m * Float64(y / z));
	end
	return Float64(x_s * tmp)
end

MATLAB

x\_m = abs(x);
x\_s = sign(x) * abs(1.0);
function tmp_2 = code(x_s, x_m, y, z, t)
	tmp = 0.0;
	if (y <= -4.5e-141)
		tmp = y * (x_m / z);
	elseif (y <= 6e-142)
		tmp = x_m * (t / z);
	else
		tmp = x_m * (y / z);
	end
	tmp_2 = x_s * tmp;
end

Wolfram

x\_m = N[Abs[x], $MachinePrecision]
x\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[x]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[x$95$s_, x$95$m_, y_, z_, t_] := N[(x$95$s * If[LessEqual[y, -4.5e-141], N[(y * N[(x$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6e-142], N[(x$95$m * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(x$95$m * N[(y / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if y < -4.5e-141

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf 87.2%

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

      1. associate-*r/ 87.2%

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

      2. associate-*r* 87.2%

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

      3. neg-mul-1 87.2%

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

      4. *-commutative 87.2%

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

      5. times-frac 87.1%

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

      6. distribute-frac-neg 87.1%

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

      7. distribute-neg-frac2 87.1%

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

      8. neg-sub0 87.1%

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

      9. associate--r- 87.1%

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

      10. metadata-eval 87.1%

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

   5. Simplified 87.1%

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

   6. Taylor expanded in t around 0 69.8%

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

   if -4.5e-141 < y < 6.0000000000000002e-142

   1. Initial program 98.2%

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

      1. clear-num 98.0%

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

      2. associate-/r/ 98.1%

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

   4. Applied egg-rr 98.1%

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

   5. Taylor expanded in z around inf 55.9%

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

      1. associate-/l* 60.5%

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

      2. neg-mul-1 60.5%

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

      3. sub-neg 60.5%

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

      4. remove-double-neg 60.5%

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

   7. Simplified 60.5%

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

   8. Taylor expanded in y around 0 58.9%

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

   if 6.0000000000000002e-142 < y

   1. Initial program 95.8%

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

   3. Taylor expanded in y around inf 75.8%

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

      1. associate-*r/ 74.0%

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

   5. Simplified 74.0%

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

Alternative 12: 24.1% accurate, 1.2× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) x)
(FPCore (x_s x_m y z t)
 :precision binary64
 (* x_s (if (<= z 1.0) (* x_m (- t)) (* 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 <= 1.0) {
		tmp = x_m * -t;
	} 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 <= 1.0d0) then
        tmp = x_m * -t
    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 <= 1.0) {
		tmp = x_m * -t;
	} 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 <= 1.0:
		tmp = x_m * -t
	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 <= 1.0)
		tmp = Float64(x_m * Float64(-t));
	else
		tmp = Float64(x_m * 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 <= 1.0)
		tmp = x_m * -t;
	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[LessEqual[z, 1.0], N[(x$95$m * (-t)), $MachinePrecision], N[(x$95$m * t), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if z < 1

   1. Initial program 95.6%

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

   3. Taylor expanded in z around 0 81.6%

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

   4. Taylor expanded in y around 0 31.5%

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

      1. neg-mul-1 31.5%

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

      2. *-commutative 31.5%

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

      3. distribute-rgt-neg-out 31.5%

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

   6. Simplified 31.5%

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

   if 1 < z

   1. Initial program 96.4%

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

   3. Taylor expanded in z around 0 28.4%

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

   4. Taylor expanded in y around 0 6.6%

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

      1. neg-mul-1 6.6%

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

      2. *-commutative 6.6%

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

      3. distribute-rgt-neg-out 6.6%

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

   6. Simplified 6.6%

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

      1. add-sqr-sqrt 3.6%

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

      2. sqrt-unprod 12.7%

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

      3. sqr-neg 12.7%

         \[\leadsto x \cdot \sqrt{\color{blue}{t \cdot t}} \]

      4. sqrt-unprod 7.8%

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

      5. add-sqr-sqrt 13.8%

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

      6. pow1 13.8%

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

   8. Applied egg-rr 13.8%

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

      1. unpow1 13.8%

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

   10. Simplified 13.8%

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

Alternative 13: 9.6% accurate, 3.7× speedup

Math

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

FPCore

x\_m = (fabs.f64 x)
x\_s = (copysign.f64 #s(literal 1 binary64) 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 * 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 68.9%

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

4. Taylor expanded in y around 0 25.6%

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

   1. neg-mul-1 25.6%

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

   2. *-commutative 25.6%

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

   3. distribute-rgt-neg-out 25.6%

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

6. Simplified 25.6%

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

   1. add-sqr-sqrt 10.7%

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

   2. sqrt-unprod 12.7%

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

   3. sqr-neg 12.7%

      \[\leadsto x \cdot \sqrt{\color{blue}{t \cdot t}} \]

   4. sqrt-unprod 3.4%

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

   5. add-sqr-sqrt 7.5%

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

   6. pow1 7.5%

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

8. Applied egg-rr 7.5%

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

   1. unpow1 7.5%

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

10. Simplified 7.5%

    \[\leadsto \color{blue}{x \cdot t} \]
11. Add Preprocessing

Developer Target 1: 95.1% 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 2024137 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< (* x (- (/ y z) (/ t (- 1 z)))) -3811613151656021/5000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (* x (- (/ y z) (* t (/ 1 (- 1 z))))) (if (< (* x (- (/ y z) (/ t (- 1 z)))) 7066972463851151/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ (/ (* y x) z) (- (/ (* t x) (- 1 z)))) (* x (- (/ y z) (* t (/ 1 (- 1 z))))))))

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