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

Percentage Accurate: 94.8% → 94.8%

Time: 9.6s

Alternatives: 13

Speedup: 1.0×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 / (z + (-1.0d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 97.1%

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

3. Final simplification 97.1%

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

Alternative 2: 94.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - \frac{t}{-z}\right)\\ \mathbf{if}\;z \leq -4.1 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) (/ t (- z))))))
   (if (<= z -4.1e+29) t_1 (if (<= z 1.0) (/ (* x (- y (* z t))) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t / -z));
	double tmp;
	if (z <= -4.1e+29) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = (x * (y - (z * t))) / 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 = x * ((y / z) - (t / -z))
    if (z <= (-4.1d+29)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = (x * (y - (z * t))) / 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 / -z));
	double tmp;
	if (z <= -4.1e+29) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = (x * (y - (z * t))) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) - (t / -z))
	tmp = 0
	if z <= -4.1e+29:
		tmp = t_1
	elif z <= 1.0:
		tmp = (x * (y - (z * t))) / 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(-z))))
	tmp = 0.0
	if (z <= -4.1e+29)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(Float64(x * Float64(y - Float64(z * t))) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) - (t / -z));
	tmp = 0.0;
	if (z <= -4.1e+29)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = (x * (y - (z * t))) / 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 / (-z)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.1e+29], t$95$1, If[LessEqual[z, 1.0], N[(N[(x * N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1000000000000003e29 or 1 < z

   1. Initial program 98.1%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if -4.1000000000000003e29 < z < 1

   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 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. *-commutative N/A

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

      16. lower-*.f64 96.5

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

   5. Applied rewrites 96.5%

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

Alternative 3: 88.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -9 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+209}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ y t) (/ x z))))
   (if (<= z -9e+33)
     t_1
     (if (<= z 1.0)
       (* x (- (/ y z) t))
       (if (<= z 8.5e+209) t_1 (/ (* x (+ y t)) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y + t) * (x / z);
	double tmp;
	if (z <= -9e+33) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} else if (z <= 8.5e+209) {
		tmp = t_1;
	} else {
		tmp = (x * (y + 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) :: t_1
    real(8) :: tmp
    t_1 = (y + t) * (x / z)
    if (z <= (-9d+33)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = x * ((y / z) - t)
    else if (z <= 8.5d+209) then
        tmp = t_1
    else
        tmp = (x * (y + t)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y + t) * (x / z);
	double tmp;
	if (z <= -9e+33) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} else if (z <= 8.5e+209) {
		tmp = t_1;
	} else {
		tmp = (x * (y + t)) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y + t) * (x / z)
	tmp = 0
	if z <= -9e+33:
		tmp = t_1
	elif z <= 1.0:
		tmp = x * ((y / z) - t)
	elif z <= 8.5e+209:
		tmp = t_1
	else:
		tmp = (x * (y + t)) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y + t) * Float64(x / z))
	tmp = 0.0
	if (z <= -9e+33)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	elseif (z <= 8.5e+209)
		tmp = t_1;
	else
		tmp = Float64(Float64(x * Float64(y + t)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y + t) * (x / z);
	tmp = 0.0;
	if (z <= -9e+33)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = x * ((y / z) - t);
	elseif (z <= 8.5e+209)
		tmp = t_1;
	else
		tmp = (x * (y + t)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9e+33], t$95$1, If[LessEqual[z, 1.0], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.5e+209], t$95$1, N[(N[(x * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.0000000000000001e33 or 1 < z < 8.50000000000000062e209

   1. Initial program 98.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 49.2

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

   5. Applied rewrites 49.2%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 89.3

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

   8. Applied rewrites 89.3%

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

   if -9.0000000000000001e33 < z < 1

   1. Initial program 96.2%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 32.6

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

   5. Applied rewrites 32.6%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 95.7

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

   8. Applied rewrites 95.7%

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

   if 8.50000000000000062e209 < z

   1. Initial program 94.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 99.7%

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

Alternative 4: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+209}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(y + t\right)}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ y t) (/ x z))))
   (if (<= z -1.4e-131)
     t_1
     (if (<= z 2.7e-13)
       (/ (* x y) z)
       (if (<= z 8.5e+209) t_1 (/ (* x (+ y t)) z))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y + t) * (x / z);
	double tmp;
	if (z <= -1.4e-131) {
		tmp = t_1;
	} else if (z <= 2.7e-13) {
		tmp = (x * y) / z;
	} else if (z <= 8.5e+209) {
		tmp = t_1;
	} else {
		tmp = (x * (y + 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) :: t_1
    real(8) :: tmp
    t_1 = (y + t) * (x / z)
    if (z <= (-1.4d-131)) then
        tmp = t_1
    else if (z <= 2.7d-13) then
        tmp = (x * y) / z
    else if (z <= 8.5d+209) then
        tmp = t_1
    else
        tmp = (x * (y + t)) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y + t) * (x / z);
	double tmp;
	if (z <= -1.4e-131) {
		tmp = t_1;
	} else if (z <= 2.7e-13) {
		tmp = (x * y) / z;
	} else if (z <= 8.5e+209) {
		tmp = t_1;
	} else {
		tmp = (x * (y + t)) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y + t) * (x / z)
	tmp = 0
	if z <= -1.4e-131:
		tmp = t_1
	elif z <= 2.7e-13:
		tmp = (x * y) / z
	elif z <= 8.5e+209:
		tmp = t_1
	else:
		tmp = (x * (y + t)) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y + t) * Float64(x / z))
	tmp = 0.0
	if (z <= -1.4e-131)
		tmp = t_1;
	elseif (z <= 2.7e-13)
		tmp = Float64(Float64(x * y) / z);
	elseif (z <= 8.5e+209)
		tmp = t_1;
	else
		tmp = Float64(Float64(x * Float64(y + t)) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y + t) * (x / z);
	tmp = 0.0;
	if (z <= -1.4e-131)
		tmp = t_1;
	elseif (z <= 2.7e-13)
		tmp = (x * y) / z;
	elseif (z <= 8.5e+209)
		tmp = t_1;
	else
		tmp = (x * (y + t)) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.4e-131], t$95$1, If[LessEqual[z, 2.7e-13], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 8.5e+209], t$95$1, N[(N[(x * N[(y + t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.4e-131 or 2.70000000000000011e-13 < z < 8.50000000000000062e209

   1. Initial program 98.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.0

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

   5. Applied rewrites 51.0%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 84.4

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

   8. Applied rewrites 84.4%

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

   if -1.4e-131 < z < 2.70000000000000011e-13

   1. Initial program 95.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

   if 8.50000000000000062e209 < z

   1. Initial program 94.5%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. remove-double-neg N/A

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

      3. neg-mul-1 N/A

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

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

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

      5. neg-mul-1 N/A

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

      6. mul-1-neg N/A

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

      7. remove-double-neg N/A

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

      8. distribute-lft-neg-in N/A

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

      9. *-commutative N/A

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

      10. mul-1-neg N/A

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

      11. lower-/.f64 N/A

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

   5. Applied rewrites 99.7%

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

Alternative 5: 94.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -4.1 \cdot 10^{+29}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;\frac{x \cdot \left(y - z \cdot t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (+ y t) z))))
   (if (<= z -4.1e+29) t_1 (if (<= z 1.0) (/ (* x (- y (* z t))) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y + t) / z);
	double tmp;
	if (z <= -4.1e+29) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = (x * (y - (z * t))) / 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 = x * ((y + t) / z)
    if (z <= (-4.1d+29)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = (x * (y - (z * t))) / 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 + t) / z);
	double tmp;
	if (z <= -4.1e+29) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = (x * (y - (z * t))) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y + t) / z)
	tmp = 0
	if z <= -4.1e+29:
		tmp = t_1
	elif z <= 1.0:
		tmp = (x * (y - (z * t))) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y + t) / z))
	tmp = 0.0
	if (z <= -4.1e+29)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(Float64(x * Float64(y - Float64(z * t))) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y + t) / z);
	tmp = 0.0;
	if (z <= -4.1e+29)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = (x * (y - (z * t))) / 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 + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -4.1e+29], t$95$1, If[LessEqual[z, 1.0], N[(N[(x * N[(y - N[(z * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.1000000000000003e29 or 1 < z

   1. Initial program 98.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. lower-+.f64 96.9

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

   5. Applied rewrites 96.9%

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

   if -4.1000000000000003e29 < z < 1

   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 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. *-commutative N/A

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

      16. lower-*.f64 96.5

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

   5. Applied rewrites 96.5%

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

Alternative 6: 93.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -9 \cdot 10^{+33}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ (+ y t) z))))
   (if (<= z -9e+33) t_1 (if (<= z 1.0) (* x (- (/ y z) t)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y + t) / z);
	double tmp;
	if (z <= -9e+33) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} 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 = x * ((y + t) / z)
    if (z <= (-9d+33)) then
        tmp = t_1
    else if (z <= 1.0d0) then
        tmp = x * ((y / z) - t)
    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 + t) / z);
	double tmp;
	if (z <= -9e+33) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y + t) / z)
	tmp = 0
	if z <= -9e+33:
		tmp = t_1
	elif z <= 1.0:
		tmp = x * ((y / z) - t)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y + t) / z))
	tmp = 0.0
	if (z <= -9e+33)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y + t) / z);
	tmp = 0.0;
	if (z <= -9e+33)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = x * ((y / z) - t);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9e+33], t$95$1, If[LessEqual[z, 1.0], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.0000000000000001e33 or 1 < z

   1. Initial program 98.1%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. lower-+.f64 96.8

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

   5. Applied rewrites 96.8%

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

   if -9.0000000000000001e33 < z < 1

   1. Initial program 96.2%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 32.6

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

   5. Applied rewrites 32.6%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. div-sub N/A

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

      4. associate-/l* N/A

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

      5. *-lft-identity N/A

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

      6. associate-*l/ N/A

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

      7. lft-mult-inverse N/A

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

      8. *-rgt-identity N/A

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

      9. lower--.f64 N/A

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

      10. lower-/.f64 95.7

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

   8. Applied rewrites 95.7%

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

Alternative 7: 78.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y + t\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{-131}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{-13}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ y t) (/ x z))))
   (if (<= z -1.4e-131) t_1 (if (<= z 2.7e-13) (/ (* x y) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y + t) * (x / z);
	double tmp;
	if (z <= -1.4e-131) {
		tmp = t_1;
	} else if (z <= 2.7e-13) {
		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 = (y + t) * (x / z)
    if (z <= (-1.4d-131)) then
        tmp = t_1
    else if (z <= 2.7d-13) 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 = (y + t) * (x / z);
	double tmp;
	if (z <= -1.4e-131) {
		tmp = t_1;
	} else if (z <= 2.7e-13) {
		tmp = (x * y) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y + t) * (x / z)
	tmp = 0
	if z <= -1.4e-131:
		tmp = t_1
	elif z <= 2.7e-13:
		tmp = (x * y) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y + t) * Float64(x / z))
	tmp = 0.0
	if (z <= -1.4e-131)
		tmp = t_1;
	elseif (z <= 2.7e-13)
		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 = (y + t) * (x / z);
	tmp = 0.0;
	if (z <= -1.4e-131)
		tmp = t_1;
	elseif (z <= 2.7e-13)
		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[(y + t), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.4e-131], t$95$1, If[LessEqual[z, 2.7e-13], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4e-131 or 2.70000000000000011e-13 < z

   1. Initial program 98.4%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 51.1

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

   5. Applied rewrites 51.1%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 82.8

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

   8. Applied rewrites 82.8%

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

   if -1.4e-131 < z < 2.70000000000000011e-13

   1. Initial program 95.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.4

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

   5. Applied rewrites 79.4%

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

Alternative 8: 66.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8.8 \cdot 10^{+86}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+146}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -8.8e+86)
   (* x (/ t z))
   (if (<= t 9.5e+146) (* x (/ y z)) (* x (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -8.8e+86) {
		tmp = x * (t / z);
	} else if (t <= 9.5e+146) {
		tmp = x * (y / 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 (t <= (-8.8d+86)) then
        tmp = x * (t / z)
    else if (t <= 9.5d+146) then
        tmp = x * (y / 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 (t <= -8.8e+86) {
		tmp = x * (t / z);
	} else if (t <= 9.5e+146) {
		tmp = x * (y / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -8.8e+86:
		tmp = x * (t / z)
	elif t <= 9.5e+146:
		tmp = x * (y / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -8.8e+86)
		tmp = Float64(x * Float64(t / z));
	elseif (t <= 9.5e+146)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -8.8e+86)
		tmp = x * (t / z);
	elseif (t <= 9.5e+146)
		tmp = x * (y / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -8.8e+86], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e+146], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.80000000000000013e86

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.3%

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

   if -8.80000000000000013e86 < t < 9.49999999999999926e146

   1. Initial program 95.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 81.2

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

   5. Applied rewrites 81.2%

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

   if 9.49999999999999926e146 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.3%

      \[\leadsto x \cdot \left(-t\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 9: 65.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{+87}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.22e+87)
   (* x (/ t z))
   (if (<= t 9.5e+146) (/ (* x y) z) (* x (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.22e+87) {
		tmp = x * (t / z);
	} else if (t <= 9.5e+146) {
		tmp = (x * y) / 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 (t <= (-1.22d+87)) then
        tmp = x * (t / z)
    else if (t <= 9.5d+146) then
        tmp = (x * y) / 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 (t <= -1.22e+87) {
		tmp = x * (t / z);
	} else if (t <= 9.5e+146) {
		tmp = (x * y) / z;
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.22e+87:
		tmp = x * (t / z)
	elif t <= 9.5e+146:
		tmp = (x * y) / z
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.22e+87)
		tmp = Float64(x * Float64(t / z));
	elseif (t <= 9.5e+146)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.22e+87)
		tmp = x * (t / z);
	elseif (t <= 9.5e+146)
		tmp = (x * y) / z;
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.22e+87], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 9.5e+146], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2200000000000001e87

   1. Initial program 99.7%

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

   3. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 89.1

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

   5. Applied rewrites 89.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.3%

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

   if -1.2200000000000001e87 < t < 9.49999999999999926e146

   1. Initial program 95.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if 9.49999999999999926e146 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.3%

      \[\leadsto x \cdot \left(-t\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 10: 64.3% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{+87}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{elif}\;t \leq 9.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{x \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.22e+87)
   (/ (* x t) z)
   (if (<= t 9.5e+146) (/ (* x y) z) (* x (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.22e+87) {
		tmp = (x * t) / z;
	} else if (t <= 9.5e+146) {
		tmp = (x * y) / 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 (t <= (-1.22d+87)) then
        tmp = (x * t) / z
    else if (t <= 9.5d+146) then
        tmp = (x * y) / 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 (t <= -1.22e+87) {
		tmp = (x * t) / z;
	} else if (t <= 9.5e+146) {
		tmp = (x * y) / z;
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.22e+87:
		tmp = (x * t) / z
	elif t <= 9.5e+146:
		tmp = (x * y) / z
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.22e+87)
		tmp = Float64(Float64(x * t) / z);
	elseif (t <= 9.5e+146)
		tmp = Float64(Float64(x * y) / z);
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.22e+87)
		tmp = (x * t) / z;
	elseif (t <= 9.5e+146)
		tmp = (x * y) / z;
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.22e+87], N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 9.5e+146], N[(N[(x * y), $MachinePrecision] / z), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2200000000000001e87

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 19.6

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

   5. Applied rewrites 19.6%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 70.1

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

   8. Applied rewrites 70.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 58.2%

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

   if -1.2200000000000001e87 < t < 9.49999999999999926e146

   1. Initial program 95.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.3

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

   5. Applied rewrites 79.3%

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

   if 9.49999999999999926e146 < t

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 55.3%

      \[\leadsto x \cdot \left(-t\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 11: 64.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{+87}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+244}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.22e+87)
   (/ (* x t) z)
   (if (<= t 1.7e+244) (* y (/ x z)) (* x (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.22e+87) {
		tmp = (x * t) / z;
	} else if (t <= 1.7e+244) {
		tmp = y * (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 (t <= (-1.22d+87)) then
        tmp = (x * t) / z
    else if (t <= 1.7d+244) then
        tmp = y * (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 (t <= -1.22e+87) {
		tmp = (x * t) / z;
	} else if (t <= 1.7e+244) {
		tmp = y * (x / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.22e+87:
		tmp = (x * t) / z
	elif t <= 1.7e+244:
		tmp = y * (x / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.22e+87)
		tmp = Float64(Float64(x * t) / z);
	elseif (t <= 1.7e+244)
		tmp = Float64(y * 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 (t <= -1.22e+87)
		tmp = (x * t) / z;
	elseif (t <= 1.7e+244)
		tmp = y * (x / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.22e+87], N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[t, 1.7e+244], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.2200000000000001e87

   1. Initial program 99.7%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 19.6

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

   5. Applied rewrites 19.6%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 70.1

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

   8. Applied rewrites 70.1%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 58.2%

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

   if -1.2200000000000001e87 < t < 1.70000000000000005e244

   1. Initial program 96.3%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 75.3

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

   5. Applied rewrites 75.3%

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

   7. Applied rewrites 72.8%

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

   if 1.70000000000000005e244 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. mul-1-neg N/A

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

      3. unsub-neg N/A

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

      4. lower--.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 67.6

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

   5. Applied rewrites 67.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 68.1%

      \[\leadsto x \cdot \left(-t\right) \]
3. Recombined 3 regimes into one program.

4. Final simplification 69.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.22 \cdot 10^{+87}:\\ \;\;\;\;\frac{x \cdot t}{z}\\ \mathbf{elif}\;t \leq 1.7 \cdot 10^{+244}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 42.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x \cdot t}{z}\\ \mathbf{if}\;z \leq -0.75:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1:\\ \;\;\;\;x \cdot \left(-\mathsf{fma}\left(z, t, t\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (* x t) z)))
   (if (<= z -0.75) t_1 (if (<= z 1.0) (* x (- (fma z t t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * t) / z;
	double tmp;
	if (z <= -0.75) {
		tmp = t_1;
	} else if (z <= 1.0) {
		tmp = x * -fma(z, t, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * t) / z)
	tmp = 0.0
	if (z <= -0.75)
		tmp = t_1;
	elseif (z <= 1.0)
		tmp = Float64(x * Float64(-fma(z, t, t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * t), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -0.75], t$95$1, If[LessEqual[z, 1.0], N[(x * (-N[(z * t + t), $MachinePrecision])), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -0.75 or 1 < z

   1. Initial program 98.2%

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

   3. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 50.2

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

   5. Applied rewrites 50.2%

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

   6. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. +-commutative N/A

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

      9. lower-+.f64 N/A

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

      10. lower-/.f64 87.0

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

   8. Applied rewrites 87.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 51.2%

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

   if -0.75 < z < 1

   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

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. mul-1-neg N/A

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

      9. remove-double-neg N/A

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

      10. metadata-eval N/A

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

      11. lower-+.f64 32.7

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

   5. Applied rewrites 32.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 32.5%

      \[\leadsto x \cdot \left(-\mathsf{fma}\left(z, t, t\right)\right) \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 22.9% accurate, 4.3× 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 97.1%

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

3. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. mul-1-neg N/A

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

   3. unsub-neg N/A

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

   4. lower--.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 63.9

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

5. Applied rewrites 63.9%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 21.6%

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

Developer Target 1: 95.2% 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 2024238 
(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)))))