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

Percentage Accurate: 93.8% → 97.2%

Time: 8.1s

Alternatives: 10

Speedup: 0.2×

• 20.2% 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: 93.8% 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: 97.2% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} + \frac{t}{z + -1}\\ t_2 := t\_1 \cdot x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-172}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 10^{-323}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;t\_1 \leq 10^{+299}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ y z) (/ t (+ z -1.0)))) (t_2 (* t_1 x)))
   (if (<= t_1 -2e-172)
     t_2
     (if (<= t_1 1e-323)
       (/ (* x (+ y t)) z)
       (if (<= t_1 1e+299) t_2 (/ 1.0 (/ z (* y x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) + (t / (z + -1.0));
	double t_2 = t_1 * x;
	double tmp;
	if (t_1 <= -2e-172) {
		tmp = t_2;
	} else if (t_1 <= 1e-323) {
		tmp = (x * (y + t)) / z;
	} else if (t_1 <= 1e+299) {
		tmp = t_2;
	} else {
		tmp = 1.0 / (z / (y * x));
	}
	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 = (y / z) + (t / (z + (-1.0d0)))
    t_2 = t_1 * x
    if (t_1 <= (-2d-172)) then
        tmp = t_2
    else if (t_1 <= 1d-323) then
        tmp = (x * (y + t)) / z
    else if (t_1 <= 1d+299) then
        tmp = t_2
    else
        tmp = 1.0d0 / (z / (y * x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / z) + (t / (z + -1.0));
	double t_2 = t_1 * x;
	double tmp;
	if (t_1 <= -2e-172) {
		tmp = t_2;
	} else if (t_1 <= 1e-323) {
		tmp = (x * (y + t)) / z;
	} else if (t_1 <= 1e+299) {
		tmp = t_2;
	} else {
		tmp = 1.0 / (z / (y * x));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) + (t / (z + -1.0))
	t_2 = t_1 * x
	tmp = 0
	if t_1 <= -2e-172:
		tmp = t_2
	elif t_1 <= 1e-323:
		tmp = (x * (y + t)) / z
	elif t_1 <= 1e+299:
		tmp = t_2
	else:
		tmp = 1.0 / (z / (y * x))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) + Float64(t / Float64(z + -1.0)))
	t_2 = Float64(t_1 * x)
	tmp = 0.0
	if (t_1 <= -2e-172)
		tmp = t_2;
	elseif (t_1 <= 1e-323)
		tmp = Float64(Float64(x * Float64(y + t)) / z);
	elseif (t_1 <= 1e+299)
		tmp = t_2;
	else
		tmp = Float64(1.0 / Float64(z / Float64(y * x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) + (t / (z + -1.0));
	t_2 = t_1 * x;
	tmp = 0.0;
	if (t_1 <= -2e-172)
		tmp = t_2;
	elseif (t_1 <= 1e-323)
		tmp = (x * (y + t)) / z;
	elseif (t_1 <= 1e+299)
		tmp = t_2;
	else
		tmp = 1.0 / (z / (y * x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] + N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-172], t$95$2, If[LessEqual[t$95$1, 1e-323], N[(N[(x * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t$95$1, 1e+299], t$95$2, N[(1.0 / N[(z / N[(y * x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -2.0000000000000001e-172 or 9.88131e-324 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 1.0000000000000001e299

   1. Initial program 98.5%

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

   if -2.0000000000000001e-172 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 9.88131e-324

   1. Initial program 61.5%

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

   3. Taylor expanded in z around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. sub-neg 99.9%

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

      3. remove-double-neg 99.9%

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

      4. neg-mul-1 99.9%

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

      5. distribute-neg-in 99.9%

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

      6. neg-mul-1 99.9%

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

      7. sub-neg 99.9%

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

      8. distribute-lft-neg-in 99.9%

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

      9. *-commutative 99.9%

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

      10. distribute-neg-frac 99.9%

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

      11. associate-/l* 61.5%

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

      12. distribute-rgt-neg-in 61.5%

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

      13. distribute-neg-frac 61.5%

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

   5. Simplified 61.5%

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

   6. Taylor expanded in t around 0 61.5%

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

      1. +-commutative 61.5%

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

   8. Simplified 61.5%

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

      1. div-inv 61.5%

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

      2. div-inv 61.5%

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

      3. distribute-rgt-out 61.5%

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

      4. associate-/r/ 55.6%

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

      5. div-inv 55.6%

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

      6. associate-/r/ 95.9%

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

      7. associate-*l/ 99.9%

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

   10. Applied egg-rr 99.9%

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

   if 1.0000000000000001e299 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 62.4%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. associate-*r/ 62.4%

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

   5. Simplified 62.4%

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

      1. associate-*r/ 99.8%

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

      2. clear-num 99.9%

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

   7. Applied egg-rr 99.9%

      \[\leadsto \color{blue}{\frac{1}{\frac{z}{x \cdot y}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} + \frac{t}{z + -1} \leq -2 \cdot 10^{-172}:\\ \;\;\;\;\left(\frac{y}{z} + \frac{t}{z + -1}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} + \frac{t}{z + -1} \leq 10^{-323}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \mathbf{elif}\;\frac{y}{z} + \frac{t}{z + -1} \leq 10^{+299}:\\ \;\;\;\;\left(\frac{y}{z} + \frac{t}{z + -1}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{z}{y \cdot x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 73.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.85e+79)
   (* (/ y z) x)
   (if (<= z 1.08e+41)
     (* x (- (/ y z) t))
     (if (<= z 6.6e+104) (* x (/ t z)) (/ (* y x) z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.85e+79) {
		tmp = (y / z) * x;
	} else if (z <= 1.08e+41) {
		tmp = x * ((y / z) - t);
	} else if (z <= 6.6e+104) {
		tmp = x * (t / z);
	} else {
		tmp = (y * x) / z;
	}
	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) :: tmp
    if (z <= (-2.85d+79)) then
        tmp = (y / z) * x
    else if (z <= 1.08d+41) then
        tmp = x * ((y / z) - t)
    else if (z <= 6.6d+104) then
        tmp = x * (t / z)
    else
        tmp = (y * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.85e+79) {
		tmp = (y / z) * x;
	} else if (z <= 1.08e+41) {
		tmp = x * ((y / z) - t);
	} else if (z <= 6.6e+104) {
		tmp = x * (t / z);
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.85e+79:
		tmp = (y / z) * x
	elif z <= 1.08e+41:
		tmp = x * ((y / z) - t)
	elif z <= 6.6e+104:
		tmp = x * (t / z)
	else:
		tmp = (y * x) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.85e+79)
		tmp = Float64(Float64(y / z) * x);
	elseif (z <= 1.08e+41)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	elseif (z <= 6.6e+104)
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(Float64(y * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.85e+79)
		tmp = (y / z) * x;
	elseif (z <= 1.08e+41)
		tmp = x * ((y / z) - t);
	elseif (z <= 6.6e+104)
		tmp = x * (t / z);
	else
		tmp = (y * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.85e+79], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.08e+41], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 6.6e+104], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.8499999999999998e79

   1. Initial program 86.5%

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

   3. Taylor expanded in y around inf 55.0%

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

      1. associate-*r/ 62.4%

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

   5. Simplified 62.4%

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

   if -2.8499999999999998e79 < z < 1.08000000000000004e41

   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 85.0%

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

      1. mul-1-neg 85.0%

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

      2. unsub-neg 85.0%

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

      3. div-sub 85.0%

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

      4. associate-/l* 85.0%

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

      5. *-inverses 85.0%

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

      6. *-rgt-identity 85.0%

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

   5. Simplified 85.0%

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

   if 1.08000000000000004e41 < z < 6.59999999999999969e104

   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 inf 94.2%

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

      1. *-commutative 94.2%

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

      2. sub-neg 94.2%

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

      3. remove-double-neg 94.2%

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

      4. neg-mul-1 94.2%

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

      5. distribute-neg-in 94.2%

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

      6. neg-mul-1 94.2%

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

      7. sub-neg 94.2%

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

      8. distribute-lft-neg-in 94.2%

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

      9. *-commutative 94.2%

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

      10. distribute-neg-frac 94.2%

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

      11. associate-/l* 99.8%

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

      12. distribute-rgt-neg-in 99.8%

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

      13. distribute-neg-frac 99.8%

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

   5. Simplified 99.8%

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

   6. Taylor expanded in t around inf 71.8%

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

   if 6.59999999999999969e104 < z

   1. Initial program 93.0%

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

   3. Taylor expanded in y around inf 66.2%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+79}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1.08 \cdot 10^{+41}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+104}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.0)
   (* x (/ (+ y t) z))
   (if (<= z 1.0) (* x (- (/ y z) t)) (* x (+ (/ y z) (/ t z))))))

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 90.9%

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

   3. Taylor expanded in z around inf 85.6%

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

      1. *-commutative 85.6%

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

      2. sub-neg 85.6%

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

      3. remove-double-neg 85.6%

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

      4. neg-mul-1 85.6%

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

      5. distribute-neg-in 85.6%

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

      6. neg-mul-1 85.6%

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

      7. sub-neg 85.6%

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

      8. distribute-lft-neg-in 85.6%

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

      9. *-commutative 85.6%

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

      10. distribute-neg-frac 85.6%

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

      11. associate-/l* 90.4%

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

      12. distribute-rgt-neg-in 90.4%

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

      13. distribute-neg-frac 90.4%

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

   5. Simplified 90.4%

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

   if -1 < z < 1

   1. Initial program 92.8%

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

   3. Taylor expanded in z around 0 91.7%

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

      1. mul-1-neg 91.7%

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

      2. unsub-neg 91.7%

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

      3. div-sub 91.7%

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

      4. associate-/l* 91.8%

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

      5. *-inverses 91.8%

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

      6. *-rgt-identity 91.8%

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

   5. Simplified 91.8%

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

   if 1 < z

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf 88.4%

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

      1. *-commutative 88.4%

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

      2. sub-neg 88.4%

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

      3. remove-double-neg 88.4%

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

      4. neg-mul-1 88.4%

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

      5. distribute-neg-in 88.4%

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

      6. neg-mul-1 88.4%

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

      7. sub-neg 88.4%

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

      8. distribute-lft-neg-in 88.4%

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

      9. *-commutative 88.4%

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

      10. distribute-neg-frac 88.4%

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

      11. associate-/l* 93.6%

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

      12. distribute-rgt-neg-in 93.6%

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

      13. distribute-neg-frac 93.6%

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

   5. Simplified 93.6%

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

   6. Taylor expanded in t around 0 93.6%

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

      1. +-commutative 93.6%

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

   8. Simplified 93.6%

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

4. Final simplification 92.0%

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

Alternative 4: 93.0% accurate, 0.6× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 1.0)))
   (* x (/ (+ y t) z))
   (* x (- (/ y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.0) || !(z <= 1.0)) {
		tmp = x * ((y + t) / z);
	} else {
		tmp = x * ((y / z) - t);
	}
	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) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = x * ((y + t) / z)
    else
        tmp = x * ((y / z) - t)
    end if
    code = tmp
end function

Java

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

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.0) or not (z <= 1.0):
		tmp = x * ((y + t) / z)
	else:
		tmp = x * ((y / z) - t)
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 93.3%

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

   3. Taylor expanded in z around inf 87.1%

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

      1. *-commutative 87.1%

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

      2. sub-neg 87.1%

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

      3. remove-double-neg 87.1%

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

      4. neg-mul-1 87.1%

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

      5. distribute-neg-in 87.1%

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

      6. neg-mul-1 87.1%

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

      7. sub-neg 87.1%

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

      8. distribute-lft-neg-in 87.1%

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

      9. *-commutative 87.1%

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

      10. distribute-neg-frac 87.1%

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

      11. associate-/l* 92.2%

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

      12. distribute-rgt-neg-in 92.2%

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

      13. distribute-neg-frac 92.2%

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

   5. Simplified 92.2%

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

   if -1 < z < 1

   1. Initial program 92.8%

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

   3. Taylor expanded in z around 0 91.7%

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

      1. mul-1-neg 91.7%

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

      2. unsub-neg 91.7%

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

      3. div-sub 91.7%

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

      4. associate-/l* 91.8%

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

      5. *-inverses 91.8%

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

      6. *-rgt-identity 91.8%

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

   5. Simplified 91.8%

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

4. Final simplification 92.0%

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

Alternative 5: 75.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+24} \lor \neg \left(t \leq 3.1 \cdot 10^{+130}\right):\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.35e+24) (not (<= t 3.1e+130)))
   (* x (/ t (+ z -1.0)))
   (* (/ y z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.35e+24) || !(t <= 3.1e+130)) {
		tmp = x * (t / (z + -1.0));
	} else {
		tmp = (y / z) * x;
	}
	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) :: tmp
    if ((t <= (-1.35d+24)) .or. (.not. (t <= 3.1d+130))) then
        tmp = x * (t / (z + (-1.0d0)))
    else
        tmp = (y / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.35e+24) || !(t <= 3.1e+130)) {
		tmp = x * (t / (z + -1.0));
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.35e+24) or not (t <= 3.1e+130):
		tmp = x * (t / (z + -1.0))
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.35e+24) || !(t <= 3.1e+130))
		tmp = Float64(x * Float64(t / Float64(z + -1.0)));
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.35e+24) || ~((t <= 3.1e+130)))
		tmp = x * (t / (z + -1.0));
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.35e+24], N[Not[LessEqual[t, 3.1e+130]], $MachinePrecision]], N[(x * N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.35e24 or 3.1e130 < t

   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 72.7%

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

      1. mul-1-neg 72.7%

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

      2. distribute-neg-frac2 72.7%

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

      3. neg-sub0 72.7%

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

      4. associate--r- 72.7%

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

      5. metadata-eval 72.7%

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

   5. Simplified 72.7%

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

   if -1.35e24 < t < 3.1e130

   1. Initial program 91.2%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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/ 80.1%

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

   5. Simplified 80.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.35 \cdot 10^{+24} \lor \neg \left(t \leq 3.1 \cdot 10^{+130}\right):\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 69.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+166} \lor \neg \left(t \leq 6 \cdot 10^{+132}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -1.15e+166) (not (<= t 6e+132))) (* x (/ t z)) (* (/ y z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.15e+166) || !(t <= 6e+132)) {
		tmp = x * (t / z);
	} else {
		tmp = (y / z) * x;
	}
	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) :: tmp
    if ((t <= (-1.15d+166)) .or. (.not. (t <= 6d+132))) then
        tmp = x * (t / z)
    else
        tmp = (y / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -1.15e+166) || !(t <= 6e+132)) {
		tmp = x * (t / z);
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -1.15e+166) or not (t <= 6e+132):
		tmp = x * (t / z)
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -1.15e+166) || !(t <= 6e+132))
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -1.15e+166) || ~((t <= 6e+132)))
		tmp = x * (t / z);
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -1.15e+166], N[Not[LessEqual[t, 6e+132]], $MachinePrecision]], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.15000000000000004e166 or 5.9999999999999996e132 < t

   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 inf 60.1%

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

      1. *-commutative 60.1%

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

      2. sub-neg 60.1%

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

      3. remove-double-neg 60.1%

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

      4. neg-mul-1 60.1%

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

      5. distribute-neg-in 60.1%

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

      6. neg-mul-1 60.1%

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

      7. sub-neg 60.1%

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

      8. distribute-lft-neg-in 60.1%

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

      9. *-commutative 60.1%

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

      10. distribute-neg-frac 60.1%

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

      11. associate-/l* 65.4%

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

      12. distribute-rgt-neg-in 65.4%

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

      13. distribute-neg-frac 65.4%

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

   5. Simplified 65.4%

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

   6. Taylor expanded in t around inf 59.0%

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

   if -1.15000000000000004e166 < t < 5.9999999999999996e132

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf 72.3%

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

      1. associate-*r/ 74.9%

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

   5. Simplified 74.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.15 \cdot 10^{+166} \lor \neg \left(t \leq 6 \cdot 10^{+132}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 41.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0001 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -0.0001) (not (<= z 1.0))) (* t (/ x z)) (* x (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.0001) || !(z <= 1.0)) {
		tmp = t * (x / z);
	} else {
		tmp = x * -t;
	}
	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) :: tmp
    if ((z <= (-0.0001d0)) .or. (.not. (z <= 1.0d0))) then
        tmp = t * (x / z)
    else
        tmp = x * -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -0.0001) || !(z <= 1.0)) {
		tmp = t * (x / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -0.0001) or not (z <= 1.0):
		tmp = t * (x / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -0.0001) || !(z <= 1.0))
		tmp = Float64(t * Float64(x / z));
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -0.0001) || ~((z <= 1.0)))
		tmp = t * (x / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -0.0001], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.00000000000000005e-4 or 1 < z

   1. Initial program 93.4%

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

   3. Taylor expanded in z around inf 87.3%

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

      1. *-commutative 87.3%

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

      2. sub-neg 87.3%

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

      3. remove-double-neg 87.3%

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

      4. neg-mul-1 87.3%

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

      5. distribute-neg-in 87.3%

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

      6. neg-mul-1 87.3%

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

      7. sub-neg 87.3%

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

      8. distribute-lft-neg-in 87.3%

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

      9. *-commutative 87.3%

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

      10. distribute-neg-frac 87.3%

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

      11. associate-/l* 92.3%

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

      12. distribute-rgt-neg-in 92.3%

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

      13. distribute-neg-frac 92.3%

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

   5. Simplified 92.3%

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

   6. Taylor expanded in t around inf 49.7%

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

      1. associate-/l* 49.0%

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

   8. Simplified 49.0%

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

   if -1.00000000000000005e-4 < z < 1

   1. Initial program 92.6%

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

   3. Taylor expanded in z around 0 91.6%

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

      1. mul-1-neg 91.6%

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

      2. unsub-neg 91.6%

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

      3. div-sub 91.6%

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

      4. associate-/l* 91.6%

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

      5. *-inverses 91.6%

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

      6. *-rgt-identity 91.6%

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

   5. Simplified 91.6%

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

   6. Taylor expanded in y around 0 31.8%

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

      1. associate-*r* 31.8%

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

      2. mul-1-neg 31.8%

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

   8. Simplified 31.8%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0001 \lor \neg \left(z \leq 1\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+166}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+129}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.2e+166)
   (* x (/ t z))
   (if (<= t 2.9e+129) (* (/ y z) x) (/ x (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.2e+166) {
		tmp = x * (t / z);
	} else if (t <= 2.9e+129) {
		tmp = (y / z) * x;
	} else {
		tmp = x / (z / t);
	}
	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) :: tmp
    if (t <= (-1.2d+166)) then
        tmp = x * (t / z)
    else if (t <= 2.9d+129) then
        tmp = (y / z) * x
    else
        tmp = x / (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.2e+166) {
		tmp = x * (t / z);
	} else if (t <= 2.9e+129) {
		tmp = (y / z) * x;
	} else {
		tmp = x / (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.2e+166:
		tmp = x * (t / z)
	elif t <= 2.9e+129:
		tmp = (y / z) * x
	else:
		tmp = x / (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.2e+166)
		tmp = Float64(x * Float64(t / z));
	elseif (t <= 2.9e+129)
		tmp = Float64(Float64(y / z) * x);
	else
		tmp = Float64(x / Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.2e+166)
		tmp = x * (t / z);
	elseif (t <= 2.9e+129)
		tmp = (y / z) * x;
	else
		tmp = x / (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.2e+166], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 2.9e+129], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], N[(x / N[(z / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.19999999999999996e166

   1. Initial program 94.0%

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

   3. Taylor expanded in z around inf 60.8%

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

      1. *-commutative 60.8%

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

      2. sub-neg 60.8%

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

      3. remove-double-neg 60.8%

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

      4. neg-mul-1 60.8%

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

      5. distribute-neg-in 60.8%

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

      6. neg-mul-1 60.8%

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

      7. sub-neg 60.8%

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

      8. distribute-lft-neg-in 60.8%

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

      9. *-commutative 60.8%

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

      10. distribute-neg-frac 60.8%

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

      11. associate-/l* 66.3%

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

      12. distribute-rgt-neg-in 66.3%

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

      13. distribute-neg-frac 66.3%

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

   5. Simplified 66.3%

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

   6. Taylor expanded in t around inf 56.0%

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

   if -1.19999999999999996e166 < t < 2.90000000000000003e129

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf 72.3%

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

      1. associate-*r/ 74.9%

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

   5. Simplified 74.9%

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

   if 2.90000000000000003e129 < t

   1. Initial program 94.4%

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

   3. Taylor expanded in z around inf 59.4%

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

      1. *-commutative 59.4%

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

      2. sub-neg 59.4%

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

      3. remove-double-neg 59.4%

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

      4. neg-mul-1 59.4%

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

      5. distribute-neg-in 59.4%

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

      6. neg-mul-1 59.4%

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

      7. sub-neg 59.4%

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

      8. distribute-lft-neg-in 59.4%

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

      9. *-commutative 59.4%

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

      10. distribute-neg-frac 59.4%

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

      11. associate-/l* 64.6%

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

      12. distribute-rgt-neg-in 64.6%

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

      13. distribute-neg-frac 64.6%

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

   5. Simplified 64.6%

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

      1. clear-num 64.6%

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

      2. un-div-inv 64.7%

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

      3. +-commutative 64.7%

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

   7. Applied egg-rr 64.7%

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

   8. Taylor expanded in y around 0 62.0%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+166}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{+129}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 42.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0001:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.0001) (* t (/ x z)) (if (<= z 1.0) (* x (- t)) (* x (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.0001) {
		tmp = t * (x / z);
	} else if (z <= 1.0) {
		tmp = x * -t;
	} else {
		tmp = x * (t / z);
	}
	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) :: tmp
    if (z <= (-0.0001d0)) then
        tmp = t * (x / z)
    else if (z <= 1.0d0) then
        tmp = x * -t
    else
        tmp = x * (t / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.0001) {
		tmp = t * (x / z);
	} else if (z <= 1.0) {
		tmp = x * -t;
	} else {
		tmp = x * (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -0.0001:
		tmp = t * (x / z)
	elif z <= 1.0:
		tmp = x * -t
	else:
		tmp = x * (t / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.0001)
		tmp = Float64(t * Float64(x / z));
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(-t));
	else
		tmp = Float64(x * Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -0.0001)
		tmp = t * (x / z);
	elseif (z <= 1.0)
		tmp = x * -t;
	else
		tmp = x * (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.0001], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.0], N[(x * (-t)), $MachinePrecision], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.00000000000000005e-4

   1. Initial program 91.2%

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

   3. Taylor expanded in z around inf 86.0%

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

      1. *-commutative 86.0%

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

      2. sub-neg 86.0%

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

      3. remove-double-neg 86.0%

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

      4. neg-mul-1 86.0%

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

      5. distribute-neg-in 86.0%

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

      6. neg-mul-1 86.0%

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

      7. sub-neg 86.0%

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

      8. distribute-lft-neg-in 86.0%

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

      9. *-commutative 86.0%

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

      10. distribute-neg-frac 86.0%

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

      11. associate-/l* 90.7%

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

      12. distribute-rgt-neg-in 90.7%

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

      13. distribute-neg-frac 90.7%

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

   5. Simplified 90.7%

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

   6. Taylor expanded in t around inf 50.6%

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

      1. associate-/l* 51.9%

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

   8. Simplified 51.9%

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

   if -1.00000000000000005e-4 < z < 1

   1. Initial program 92.6%

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

   3. Taylor expanded in z around 0 91.6%

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

      1. mul-1-neg 91.6%

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

      2. unsub-neg 91.6%

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

      3. div-sub 91.6%

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

      4. associate-/l* 91.6%

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

      5. *-inverses 91.6%

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

      6. *-rgt-identity 91.6%

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

   5. Simplified 91.6%

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

   6. Taylor expanded in y around 0 31.8%

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

      1. associate-*r* 31.8%

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

      2. mul-1-neg 31.8%

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

   8. Simplified 31.8%

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

   if 1 < z

   1. Initial program 95.2%

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

   3. Taylor expanded in z around inf 88.4%

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

      1. *-commutative 88.4%

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

      2. sub-neg 88.4%

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

      3. remove-double-neg 88.4%

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

      4. neg-mul-1 88.4%

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

      5. distribute-neg-in 88.4%

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

      6. neg-mul-1 88.4%

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

      7. sub-neg 88.4%

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

      8. distribute-lft-neg-in 88.4%

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

      9. *-commutative 88.4%

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

      10. distribute-neg-frac 88.4%

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

      11. associate-/l* 93.6%

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

      12. distribute-rgt-neg-in 93.6%

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

      13. distribute-neg-frac 93.6%

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

   5. Simplified 93.6%

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

   6. Taylor expanded in t around inf 53.7%

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

4. Final simplification 43.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0001:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 22.0% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 93.1%

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

3. Taylor expanded in z around 0 62.3%

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

   1. mul-1-neg 62.3%

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

   2. unsub-neg 62.3%

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

   3. div-sub 62.3%

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

   4. associate-/l* 62.4%

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

   5. *-inverses 62.4%

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

   6. *-rgt-identity 62.4%

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

5. Simplified 62.4%

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

6. Taylor expanded in y around 0 21.0%

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

   1. associate-*r* 21.0%

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

   2. mul-1-neg 21.0%

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

8. Simplified 21.0%

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

9. Final simplification 21.0%

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

Developer target: 94.4% 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 2024085 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :alt
  (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)))))