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

Percentage Accurate: 94.5% → 98.3%

Time: 7.3s

Alternatives: 9

Speedup: 0.9×

• 21.1% 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: 94.5% 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: 98.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot y\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+292}:\\ \;\;\;\;x \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) y)) (t_2 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 1e+292) (* x t_2) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * y;
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 1e+292) {
		tmp = x * t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * y;
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 1e+292) {
		tmp = x * t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * y
	t_2 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 1e+292:
		tmp = x * t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * y)
	t_2 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= 1e+292)
		tmp = Float64(x * t_2);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) * y;
	t_2 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 1e+292)
		tmp = x * t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, 1e+292], N[(x * t$95$2), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < -inf.0 or 1e292 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 63.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 99.9

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

   5. Applied rewrites 99.9%

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

   7. Applied rewrites 99.9%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 1e292

   1. Initial program 97.7%

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

Alternative 2: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y}{z}\\ t_2 := \left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{if}\;y \leq -4 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-143}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-185}:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-20}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x y) z)) (t_2 (* (- (/ y z) t) x)))
   (if (<= y -4e+61)
     t_1
     (if (<= y -2.4e-143)
       t_2
       (if (<= y 3.8e-185)
         (/ (* x t) (- z 1.0))
         (if (<= y 1.18e-20) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * y) / z;
	double t_2 = ((y / z) - t) * x;
	double tmp;
	if (y <= -4e+61) {
		tmp = t_1;
	} else if (y <= -2.4e-143) {
		tmp = t_2;
	} else if (y <= 3.8e-185) {
		tmp = (x * t) / (z - 1.0);
	} else if (y <= 1.18e-20) {
		tmp = t_2;
	} 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_2 = ((y / z) - t) * x
    if (y <= (-4d+61)) then
        tmp = t_1
    else if (y <= (-2.4d-143)) then
        tmp = t_2
    else if (y <= 3.8d-185) then
        tmp = (x * t) / (z - 1.0d0)
    else if (y <= 1.18d-20) then
        tmp = t_2
    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;
	double t_2 = ((y / z) - t) * x;
	double tmp;
	if (y <= -4e+61) {
		tmp = t_1;
	} else if (y <= -2.4e-143) {
		tmp = t_2;
	} else if (y <= 3.8e-185) {
		tmp = (x * t) / (z - 1.0);
	} else if (y <= 1.18e-20) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * y) / z
	t_2 = ((y / z) - t) * x
	tmp = 0
	if y <= -4e+61:
		tmp = t_1
	elif y <= -2.4e-143:
		tmp = t_2
	elif y <= 3.8e-185:
		tmp = (x * t) / (z - 1.0)
	elif y <= 1.18e-20:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * y) / z)
	t_2 = Float64(Float64(Float64(y / z) - t) * x)
	tmp = 0.0
	if (y <= -4e+61)
		tmp = t_1;
	elseif (y <= -2.4e-143)
		tmp = t_2;
	elseif (y <= 3.8e-185)
		tmp = Float64(Float64(x * t) / Float64(z - 1.0));
	elseif (y <= 1.18e-20)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * y) / z;
	t_2 = ((y / z) - t) * x;
	tmp = 0.0;
	if (y <= -4e+61)
		tmp = t_1;
	elseif (y <= -2.4e-143)
		tmp = t_2;
	elseif (y <= 3.8e-185)
		tmp = (x * t) / (z - 1.0);
	elseif (y <= 1.18e-20)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -4e+61], t$95$1, If[LessEqual[y, -2.4e-143], t$95$2, If[LessEqual[y, 3.8e-185], N[(N[(x * t), $MachinePrecision] / N[(z - 1.0), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.18e-20], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999998e61 or 1.1800000000000001e-20 < y

   1. Initial program 88.2%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -3.9999999999999998e61 < y < -2.3999999999999999e-143 or 3.7999999999999999e-185 < y < 1.1800000000000001e-20

   1. Initial program 98.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-out-- N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 69.4

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

   5. Applied rewrites 69.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 29.3%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 73.6%

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

   if -2.3999999999999999e-143 < y < 3.7999999999999999e-185

   1. Initial program 93.9%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. sub-neg N/A

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

      6. mul-1-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. sub-neg N/A

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

      12. lower--.f64 75.8

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

   5. Applied rewrites 75.8%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+61}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-143}:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{elif}\;y \leq 3.8 \cdot 10^{-185}:\\ \;\;\;\;\frac{x \cdot t}{z - 1}\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot y}{z}\\ t_2 := \left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{if}\;y \leq -4 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-179}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-20}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x y) z)) (t_2 (* (- (/ y z) t) x)))
   (if (<= y -4e+61)
     t_1
     (if (<= y -2.1e-179)
       t_2
       (if (<= y 5.6e-226) (* (/ t z) x) (if (<= y 1.18e-20) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * y) / z;
	double t_2 = ((y / z) - t) * x;
	double tmp;
	if (y <= -4e+61) {
		tmp = t_1;
	} else if (y <= -2.1e-179) {
		tmp = t_2;
	} else if (y <= 5.6e-226) {
		tmp = (t / z) * x;
	} else if (y <= 1.18e-20) {
		tmp = t_2;
	} 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_2 = ((y / z) - t) * x
    if (y <= (-4d+61)) then
        tmp = t_1
    else if (y <= (-2.1d-179)) then
        tmp = t_2
    else if (y <= 5.6d-226) then
        tmp = (t / z) * x
    else if (y <= 1.18d-20) then
        tmp = t_2
    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;
	double t_2 = ((y / z) - t) * x;
	double tmp;
	if (y <= -4e+61) {
		tmp = t_1;
	} else if (y <= -2.1e-179) {
		tmp = t_2;
	} else if (y <= 5.6e-226) {
		tmp = (t / z) * x;
	} else if (y <= 1.18e-20) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * y) / z
	t_2 = ((y / z) - t) * x
	tmp = 0
	if y <= -4e+61:
		tmp = t_1
	elif y <= -2.1e-179:
		tmp = t_2
	elif y <= 5.6e-226:
		tmp = (t / z) * x
	elif y <= 1.18e-20:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * y) / z)
	t_2 = Float64(Float64(Float64(y / z) - t) * x)
	tmp = 0.0
	if (y <= -4e+61)
		tmp = t_1;
	elseif (y <= -2.1e-179)
		tmp = t_2;
	elseif (y <= 5.6e-226)
		tmp = Float64(Float64(t / z) * x);
	elseif (y <= 1.18e-20)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * y) / z;
	t_2 = ((y / z) - t) * x;
	tmp = 0.0;
	if (y <= -4e+61)
		tmp = t_1;
	elseif (y <= -2.1e-179)
		tmp = t_2;
	elseif (y <= 5.6e-226)
		tmp = (t / z) * x;
	elseif (y <= 1.18e-20)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[y, -4e+61], t$95$1, If[LessEqual[y, -2.1e-179], t$95$2, If[LessEqual[y, 5.6e-226], N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 1.18e-20], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.9999999999999998e61 or 1.1800000000000001e-20 < y

   1. Initial program 88.2%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 88.0

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

   5. Applied rewrites 88.0%

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

   if -3.9999999999999998e61 < y < -2.0999999999999999e-179 or 5.60000000000000016e-226 < y < 1.1800000000000001e-20

   1. Initial program 98.8%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-out-- N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 31.9%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 71.2%

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

   if -2.0999999999999999e-179 < y < 5.60000000000000016e-226

   1. Initial program 92.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. distribute-neg-in N/A

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

      21. metadata-eval N/A

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

      22. remove-double-neg N/A

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

      23. lower-+.f64 61.6

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

   4. Applied rewrites 61.6%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 75.2

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

   7. Applied rewrites 75.2%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 69.1%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4 \cdot 10^{+61}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;y \leq -2.1 \cdot 10^{-179}:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{elif}\;y \leq 5.6 \cdot 10^{-226}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;y \leq 1.18 \cdot 10^{-20}:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 95.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{-29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\left(y - t \cdot z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -6.5e-29) t_1 (if (<= z 1.0) (/ (* (- y (* t z)) x) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -6.5e-29) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = ((y - (t * z)) * x) / 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) :: tmp
    t_1 = ((t + y) / z) * x
    if (z <= (-6.5d-29)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = ((y - (t * z)) * x) / 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 = ((t + y) / z) * x;
	double tmp;
	if (z <= -6.5e-29) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = ((y - (t * z)) * x) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) / z) * x
	tmp = 0
	if z <= -6.5e-29:
		tmp = t_1
	elif z <= 1.0:
		tmp = ((y - (t * z)) * x) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -6.5e-29)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(y - Float64(t * z)) * x) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) / z) * x;
	tmp = 0.0;
	if (z <= -6.5e-29)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = ((y - (t * z)) * x) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -6.5e-29], t$95$1, If[LessEqual[z, 1.0], N[(N[(N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.5e-29 or 1 < z

   1. Initial program 96.2%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.2

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

   5. Applied rewrites 96.2%

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

   if -6.5e-29 < z < 1

   1. Initial program 88.6%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-out-- N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 96.3

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

   5. Applied rewrites 96.3%

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

4. Final simplification 96.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{-29}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{\left(y - t \cdot z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t + y}{z} \cdot x\\ \mathbf{if}\;z \leq -75000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ (+ t y) z) x)))
   (if (<= z -75000000.0) t_1 (if (<= z 1.0) (* (- (/ y z) t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t + y) / z) * x;
	double tmp;
	if (z <= -75000000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = ((y / z) - t) * x;
	} 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) :: tmp
    t_1 = ((t + y) / z) * x
    if (z <= (-75000000.0d0)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = ((y / z) - t) * x
    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 = ((t + y) / z) * x;
	double tmp;
	if (z <= -75000000.0) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = ((y / z) - t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t + y) / z) * x
	tmp = 0
	if z <= -75000000.0:
		tmp = t_1
	elif z <= 1.0:
		tmp = ((y / z) - t) * x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t + y) / z) * x)
	tmp = 0.0
	if (z <= -75000000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(Float64(Float64(y / z) - t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t + y) / z) * x;
	tmp = 0.0;
	if (z <= -75000000.0)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = ((y / z) - t) * x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t + y), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -75000000.0], t$95$1, If[LessEqual[z, 1.0], N[(N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5e7 or 1 < z

   1. Initial program 96.1%

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

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. cancel-sign-sub-inv N/A

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

      3. metadata-eval N/A

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

      4. *-lft-identity N/A

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

      5. +-commutative N/A

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

      6. lower-+.f64 96.1

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

   5. Applied rewrites 96.1%

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

   if -7.5e7 < z < 1

   1. Initial program 89.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. +-commutative N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. associate-*l* N/A

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

      8. distribute-lft-out-- N/A

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

      9. unsub-neg N/A

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

      10. mul-1-neg N/A

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

      11. lower-*.f64 N/A

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

      12. mul-1-neg N/A

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

      13. unsub-neg N/A

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

      14. lower--.f64 N/A

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

      15. lower-*.f64 96.4

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

   5. Applied rewrites 96.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 26.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 88.6%

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

4. Final simplification 92.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -75000000:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\left(\frac{y}{z} - t\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{t + y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 68.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{z} \cdot x\\ \mathbf{if}\;t \leq -4.8 \cdot 10^{+84}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ t z) x)))
   (if (<= t -4.8e+84) t_1 (if (<= t 6.8e+111) (/ (* x y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / z) * x;
	double tmp;
	if (t <= -4.8e+84) {
		tmp = t_1;
	} else if (t <= 6.8e+111) {
		tmp = (x * y) / 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) :: tmp
    t_1 = (t / z) * x
    if (t <= (-4.8d+84)) then
        tmp = t_1
    else if (t <= 6.8d+111) then
        tmp = (x * y) / 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 = (t / z) * x;
	double tmp;
	if (t <= -4.8e+84) {
		tmp = t_1;
	} else if (t <= 6.8e+111) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / z) * x
	tmp = 0
	if t <= -4.8e+84:
		tmp = t_1
	elif t <= 6.8e+111:
		tmp = (x * y) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / z) * x)
	tmp = 0.0
	if (t <= -4.8e+84)
		tmp = t_1;
	elseif (t <= 6.8e+111)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / z) * x;
	tmp = 0.0;
	if (t <= -4.8e+84)
		tmp = t_1;
	elseif (t <= 6.8e+111)
		tmp = (x * y) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -4.8e+84], t$95$1, If[LessEqual[t, 6.8e+111], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.7999999999999999e84 or 6.8000000000000003e111 < t

   1. Initial program 92.1%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. lift-/.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. frac-add N/A

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

      7. lower-/.f64 N/A

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

      8. lower-fma.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. distribute-neg-in N/A

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

      12. metadata-eval N/A

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

      13. remove-double-neg N/A

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

      14. lower-+.f64 N/A

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

      15. *-commutative N/A

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

      16. lower-*.f64 N/A

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

      17. lower-*.f64 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. distribute-neg-in N/A

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

      21. metadata-eval N/A

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

      22. remove-double-neg N/A

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

      23. lower-+.f64 55.4

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

   4. Applied rewrites 55.4%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-+.f64 64.7

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

   7. Applied rewrites 64.7%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 58.3%

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

   if -4.7999999999999999e84 < t < 6.8000000000000003e111

   1. Initial program 92.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 82.7

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

   5. Applied rewrites 82.7%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.8 \cdot 10^{+84}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \mathbf{elif}\;t \leq 6.8 \cdot 10^{+111}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 62.0% accurate, 2.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return (x * y) / 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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.5%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 65.0

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

5. Applied rewrites 65.0%

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

6. Final simplification 65.0%

   \[\leadsto \frac{x \cdot y}{z} \]
7. Add Preprocessing

Alternative 8: 61.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.5%

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

3. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 65.0

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

5. Applied rewrites 65.0%

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

7. Applied rewrites 60.8%

   \[\leadsto \color{blue}{\frac{x}{z} \cdot y} \]
8. Add Preprocessing

Alternative 9: 22.9% accurate, 4.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 92.5%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. +-commutative N/A

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

   3. mul-1-neg N/A

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

   4. unsub-neg N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. associate-*l* N/A

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

   8. distribute-lft-out-- N/A

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

   9. unsub-neg N/A

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

   10. mul-1-neg N/A

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

   11. lower-*.f64 N/A

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

   12. mul-1-neg N/A

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

   13. unsub-neg N/A

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

   14. lower--.f64 N/A

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

   15. lower-*.f64 67.3

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

5. Applied rewrites 67.3%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 20.2%

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

Developer Target 1: 94.9% accurate, 0.3× 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 2024249 
(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)))))