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

Percentage Accurate: 94.6% → 96.7%

Time: 8.3s

Alternatives: 12

Speedup: 0.3×

• 20.4% 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: 94.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{\frac{z}{x}}\\ t_2 := \frac{y}{z} - \frac{t}{1 - z}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+255}:\\ \;\;\;\;x \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ y (/ z x))) (t_2 (- (/ y z) (/ t (- 1.0 z)))))
   (if (<= t_2 (- INFINITY)) t_1 (if (<= t_2 4e+255) (* x t_2) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y / (z / x);
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= 4e+255) {
		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 = y / (z / x);
	double t_2 = (y / z) - (t / (1.0 - z));
	double tmp;
	if (t_2 <= -Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else if (t_2 <= 4e+255) {
		tmp = x * t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = y / (z / x)
	t_2 = (y / z) - (t / (1.0 - z))
	tmp = 0
	if t_2 <= -math.inf:
		tmp = t_1
	elif t_2 <= 4e+255:
		tmp = x * t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(y / Float64(z / x))
	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 <= 4e+255)
		tmp = Float64(x * t_2);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = y / (z / x);
	t_2 = (y / z) - (t / (1.0 - z));
	tmp = 0.0;
	if (t_2 <= -Inf)
		tmp = t_1;
	elseif (t_2 <= 4e+255)
		tmp = x * t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y / N[(z / x), $MachinePrecision]), $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, 4e+255], 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 3.99999999999999995e255 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z)))

   1. Initial program 72.6%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 65.8

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

   5. Applied rewrites 65.8%

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

   7. Applied rewrites 99.8%

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

   9. Applied rewrites 100.0%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 #s(literal 1 binary64) z))) < 3.99999999999999995e255

   1. Initial program 98.0%

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

Alternative 2: 90.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(\frac{t}{z} + y\right) + t\\ \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\frac{x}{z} \cdot t\_1\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(y - t \cdot z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_1 \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (+ (/ t z) y) t)))
   (if (<= z -130000.0)
     (* (/ x z) t_1)
     (if (<= z 6.4e-7) (/ (* (- y (* t z)) x) z) (/ (* t_1 x) z)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((t / z) + y) + t;
	double tmp;
	if (z <= -130000.0) {
		tmp = (x / z) * t_1;
	} else if (z <= 6.4e-7) {
		tmp = ((y - (t * z)) * x) / z;
	} else {
		tmp = (t_1 * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((t / z) + y) + t
    if (z <= (-130000.0d0)) then
        tmp = (x / z) * t_1
    else if (z <= 6.4d-7) then
        tmp = ((y - (t * z)) * x) / z
    else
        tmp = (t_1 * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((t / z) + y) + t;
	double tmp;
	if (z <= -130000.0) {
		tmp = (x / z) * t_1;
	} else if (z <= 6.4e-7) {
		tmp = ((y - (t * z)) * x) / z;
	} else {
		tmp = (t_1 * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((t / z) + y) + t
	tmp = 0
	if z <= -130000.0:
		tmp = (x / z) * t_1
	elif z <= 6.4e-7:
		tmp = ((y - (t * z)) * x) / z
	else:
		tmp = (t_1 * x) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(t / z) + y) + t)
	tmp = 0.0
	if (z <= -130000.0)
		tmp = Float64(Float64(x / z) * t_1);
	elseif (z <= 6.4e-7)
		tmp = Float64(Float64(Float64(y - Float64(t * z)) * x) / z);
	else
		tmp = Float64(Float64(t_1 * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((t / z) + y) + t;
	tmp = 0.0;
	if (z <= -130000.0)
		tmp = (x / z) * t_1;
	elseif (z <= 6.4e-7)
		tmp = ((y - (t * z)) * x) / z;
	else
		tmp = (t_1 * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(t / z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -130000.0], N[(N[(x / z), $MachinePrecision] * t$95$1), $MachinePrecision], If[LessEqual[z, 6.4e-7], N[(N[(N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(t$95$1 * x), $MachinePrecision] / z), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e5

   1. Initial program 96.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 82.0%

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

   7. Applied rewrites 91.5%

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

   if -1.3e5 < z < 6.4000000000000001e-7

   1. Initial program 90.9%

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

   3. Taylor expanded in z around 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 92.2

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

   5. Applied rewrites 92.2%

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

   if 6.4000000000000001e-7 < z

   1. Initial program 96.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 87.0%

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

4. Final simplification 90.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(\frac{t}{z} + y\right) + t\right)\\ \mathbf{elif}\;z \leq 6.4 \cdot 10^{-7}:\\ \;\;\;\;\frac{\left(y - t \cdot z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\left(\frac{t}{z} + y\right) + t\right) \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(\frac{t}{z} + y\right) + t\right)\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(y - t \cdot z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -130000.0)
   (* (/ x z) (+ (+ (/ t z) y) t))
   (if (<= z 2.45e-18) (/ (* (- y (* t z)) x) z) (/ (* (+ t y) x) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -130000.0) {
		tmp = (x / z) * (((t / z) + y) + t);
	} else if (z <= 2.45e-18) {
		tmp = ((y - (t * z)) * x) / z;
	} else {
		tmp = ((t + y) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-130000.0d0)) then
        tmp = (x / z) * (((t / z) + y) + t)
    else if (z <= 2.45d-18) then
        tmp = ((y - (t * z)) * x) / z
    else
        tmp = ((t + y) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -130000.0) {
		tmp = (x / z) * (((t / z) + y) + t);
	} else if (z <= 2.45e-18) {
		tmp = ((y - (t * z)) * x) / z;
	} else {
		tmp = ((t + y) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -130000.0:
		tmp = (x / z) * (((t / z) + y) + t)
	elif z <= 2.45e-18:
		tmp = ((y - (t * z)) * x) / z
	else:
		tmp = ((t + y) * x) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -130000.0)
		tmp = Float64(Float64(x / z) * Float64(Float64(Float64(t / z) + y) + t));
	elseif (z <= 2.45e-18)
		tmp = Float64(Float64(Float64(y - Float64(t * z)) * x) / z);
	else
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -130000.0)
		tmp = (x / z) * (((t / z) + y) + t);
	elseif (z <= 2.45e-18)
		tmp = ((y - (t * z)) * x) / z;
	else
		tmp = ((t + y) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -130000.0], N[(N[(x / z), $MachinePrecision] * N[(N[(N[(t / z), $MachinePrecision] + y), $MachinePrecision] + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.45e-18], N[(N[(N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e5

   1. Initial program 96.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 82.0%

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

   7. Applied rewrites 91.5%

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

   if -1.3e5 < z < 2.4500000000000001e-18

   1. Initial program 90.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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 92.2

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

   5. Applied rewrites 92.2%

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

   if 2.4500000000000001e-18 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\frac{x}{z} \cdot \left(\left(\frac{t}{z} + y\right) + t\right)\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(y - t \cdot z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\left(t + y\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(y - t \cdot z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -130000.0)
   (* (+ t y) (/ x z))
   (if (<= z 2.45e-18) (/ (* (- y (* t z)) x) z) (/ (* (+ t y) x) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -130000.0) {
		tmp = (t + y) * (x / z);
	} else if (z <= 2.45e-18) {
		tmp = ((y - (t * z)) * x) / z;
	} else {
		tmp = ((t + y) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-130000.0d0)) then
        tmp = (t + y) * (x / z)
    else if (z <= 2.45d-18) then
        tmp = ((y - (t * z)) * x) / z
    else
        tmp = ((t + y) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -130000.0) {
		tmp = (t + y) * (x / z);
	} else if (z <= 2.45e-18) {
		tmp = ((y - (t * z)) * x) / z;
	} else {
		tmp = ((t + y) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -130000.0:
		tmp = (t + y) * (x / z)
	elif z <= 2.45e-18:
		tmp = ((y - (t * z)) * x) / z
	else:
		tmp = ((t + y) * x) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -130000.0)
		tmp = Float64(Float64(t + y) * Float64(x / z));
	elseif (z <= 2.45e-18)
		tmp = Float64(Float64(Float64(y - Float64(t * z)) * x) / z);
	else
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -130000.0)
		tmp = (t + y) * (x / z);
	elseif (z <= 2.45e-18)
		tmp = ((y - (t * z)) * x) / z;
	else
		tmp = ((t + y) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -130000.0], N[(N[(t + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.45e-18], N[(N[(N[(y - N[(t * z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3e5

   1. Initial program 96.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 82.0%

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 91.5%

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

   if -1.3e5 < z < 2.4500000000000001e-18

   1. Initial program 90.8%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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 92.2

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

   5. Applied rewrites 92.2%

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

   if 2.4500000000000001e-18 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.3%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\left(t + y\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{\left(y - t \cdot z\right) \cdot x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 75.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{z - 1} \cdot x\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 490000000:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ t (- z 1.0)) x)))
   (if (<= t -3.6e+15) t_1 (if (<= t 490000000.0) (/ x (/ z y)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / (z - 1.0)) * x;
	double tmp;
	if (t <= -3.6e+15) {
		tmp = t_1;
	} else if (t <= 490000000.0) {
		tmp = x / (z / y);
	} 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 - 1.0d0)) * x
    if (t <= (-3.6d+15)) then
        tmp = t_1
    else if (t <= 490000000.0d0) then
        tmp = x / (z / y)
    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 - 1.0)) * x;
	double tmp;
	if (t <= -3.6e+15) {
		tmp = t_1;
	} else if (t <= 490000000.0) {
		tmp = x / (z / y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / (z - 1.0)) * x
	tmp = 0
	if t <= -3.6e+15:
		tmp = t_1
	elif t <= 490000000.0:
		tmp = x / (z / y)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / Float64(z - 1.0)) * x)
	tmp = 0.0
	if (t <= -3.6e+15)
		tmp = t_1;
	elseif (t <= 490000000.0)
		tmp = Float64(x / Float64(z / y));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / (z - 1.0)) * x;
	tmp = 0.0;
	if (t <= -3.6e+15)
		tmp = t_1;
	elseif (t <= 490000000.0)
		tmp = x / (z / y);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -3.6e+15], t$95$1, If[LessEqual[t, 490000000.0], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.6e15 or 4.9e8 < t

   1. Initial program 95.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. sub-neg N/A

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

      11. lower--.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -3.6e15 < t < 4.9e8

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

   7. Applied rewrites 88.1%

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

4. Final simplification 82.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{t}{z - 1} \cdot x\\ \mathbf{elif}\;t \leq 490000000:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - 1} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{z - 1} \cdot x\\ \mathbf{if}\;t \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 490000000:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ t (- z 1.0)) x)))
   (if (<= t -4.6e+15) t_1 (if (<= t 490000000.0) (* x (/ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / (z - 1.0)) * x;
	double tmp;
	if (t <= -4.6e+15) {
		tmp = t_1;
	} else if (t <= 490000000.0) {
		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 - 1.0d0)) * x
    if (t <= (-4.6d+15)) then
        tmp = t_1
    else if (t <= 490000000.0d0) 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 - 1.0)) * x;
	double tmp;
	if (t <= -4.6e+15) {
		tmp = t_1;
	} else if (t <= 490000000.0) {
		tmp = x * (y / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / (z - 1.0)) * x
	tmp = 0
	if t <= -4.6e+15:
		tmp = t_1
	elif t <= 490000000.0:
		tmp = x * (y / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / Float64(z - 1.0)) * x)
	tmp = 0.0
	if (t <= -4.6e+15)
		tmp = t_1;
	elseif (t <= 490000000.0)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / (z - 1.0)) * x;
	tmp = 0.0;
	if (t <= -4.6e+15)
		tmp = t_1;
	elseif (t <= 490000000.0)
		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 / N[(z - 1.0), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[t, -4.6e+15], t$95$1, If[LessEqual[t, 490000000.0], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.6e15 or 4.9e8 < t

   1. Initial program 95.5%

      \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
   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. sub-neg N/A

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

      11. lower--.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -4.6e15 < t < 4.9e8

   1. Initial program 92.7%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 87.5

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

   5. Applied rewrites 87.5%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{+15}:\\ \;\;\;\;\frac{t}{z - 1} \cdot x\\ \mathbf{elif}\;t \leq 490000000:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - 1} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 77.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{-151}:\\ \;\;\;\;\left(t + y\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.35e-151)
   (* (+ t y) (/ x z))
   (if (<= z 2.45e-18) (* (/ x z) y) (/ (* (+ t y) x) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.35e-151) {
		tmp = (t + y) * (x / z);
	} else if (z <= 2.45e-18) {
		tmp = (x / z) * y;
	} else {
		tmp = ((t + y) * x) / z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2.35d-151)) then
        tmp = (t + y) * (x / z)
    else if (z <= 2.45d-18) then
        tmp = (x / z) * y
    else
        tmp = ((t + y) * x) / z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.35e-151) {
		tmp = (t + y) * (x / z);
	} else if (z <= 2.45e-18) {
		tmp = (x / z) * y;
	} else {
		tmp = ((t + y) * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.35e-151:
		tmp = (t + y) * (x / z)
	elif z <= 2.45e-18:
		tmp = (x / z) * y
	else:
		tmp = ((t + y) * x) / z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.35e-151)
		tmp = Float64(Float64(t + y) * Float64(x / z));
	elseif (z <= 2.45e-18)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = Float64(Float64(Float64(t + y) * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.35e-151)
		tmp = (t + y) * (x / z);
	elseif (z <= 2.45e-18)
		tmp = (x / z) * y;
	else
		tmp = ((t + y) * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.35e-151], N[(N[(t + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.45e-18], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], N[(N[(N[(t + y), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.35000000000000014e-151

   1. Initial program 96.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 69.9%

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

   7. Applied rewrites 75.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 82.8%

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

   if -2.35000000000000014e-151 < z < 2.4500000000000001e-18

   1. Initial program 89.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 77.1%

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

   if 2.4500000000000001e-18 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 86.3%

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

4. Final simplification 81.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{-151}:\\ \;\;\;\;\left(t + y\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(t + y\right) \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.6% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t + y\right) \cdot \frac{x}{z}\\ \mathbf{if}\;z \leq -2.35 \cdot 10^{-151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (+ t y) (/ x z))))
   (if (<= z -2.35e-151) t_1 (if (<= z 2.45e-18) (* (/ x z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t + y) * (x / z);
	double tmp;
	if (z <= -2.35e-151) {
		tmp = t_1;
	} else if (z <= 2.45e-18) {
		tmp = (x / z) * y;
	} 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) * (x / z)
    if (z <= (-2.35d-151)) then
        tmp = t_1
    else if (z <= 2.45d-18) then
        tmp = (x / z) * y
    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) * (x / z);
	double tmp;
	if (z <= -2.35e-151) {
		tmp = t_1;
	} else if (z <= 2.45e-18) {
		tmp = (x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t + y) * (x / z)
	tmp = 0
	if z <= -2.35e-151:
		tmp = t_1
	elif z <= 2.45e-18:
		tmp = (x / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t + y) * Float64(x / z))
	tmp = 0.0
	if (z <= -2.35e-151)
		tmp = t_1;
	elseif (z <= 2.45e-18)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t + y) * (x / z);
	tmp = 0.0;
	if (z <= -2.35e-151)
		tmp = t_1;
	elseif (z <= 2.45e-18)
		tmp = (x / z) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t + y), $MachinePrecision] * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.35e-151], t$95$1, If[LessEqual[z, 2.45e-18], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.35000000000000014e-151 or 2.4500000000000001e-18 < z

   1. Initial program 96.7%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 77.0%

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

   7. Applied rewrites 79.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 83.6%

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

   if -2.35000000000000014e-151 < z < 2.4500000000000001e-18

   1. Initial program 89.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 67.7

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

   5. Applied rewrites 67.7%

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

   7. Applied rewrites 77.1%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.35 \cdot 10^{-151}:\\ \;\;\;\;\left(t + y\right) \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(t + y\right) \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot t\\ \mathbf{if}\;t \leq -1.05 \cdot 10^{+16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+101}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) t)))
   (if (<= t -1.05e+16) t_1 (if (<= t 2.65e+101) (* x (/ y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * t;
	double tmp;
	if (t <= -1.05e+16) {
		tmp = t_1;
	} else if (t <= 2.65e+101) {
		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 = (x / z) * t
    if (t <= (-1.05d+16)) then
        tmp = t_1
    else if (t <= 2.65d+101) 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 = (x / z) * t;
	double tmp;
	if (t <= -1.05e+16) {
		tmp = t_1;
	} else if (t <= 2.65e+101) {
		tmp = x * (y / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * t
	tmp = 0
	if t <= -1.05e+16:
		tmp = t_1
	elif t <= 2.65e+101:
		tmp = x * (y / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * t)
	tmp = 0.0
	if (t <= -1.05e+16)
		tmp = t_1;
	elseif (t <= 2.65e+101)
		tmp = Float64(x * Float64(y / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) * t;
	tmp = 0.0;
	if (t <= -1.05e+16)
		tmp = t_1;
	elseif (t <= 2.65e+101)
		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[(x / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -1.05e+16], t$95$1, If[LessEqual[t, 2.65e+101], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.05e16 or 2.65000000000000003e101 < t

   1. Initial program 95.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   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 67.9

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

   5. Applied rewrites 67.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 47.6%

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

   if -1.05e16 < t < 2.65000000000000003e101

   1. Initial program 93.0%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 83.3

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

   5. Applied rewrites 83.3%

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

4. Final simplification 70.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.05 \cdot 10^{+16}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+101}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 63.8% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot t\\ \mathbf{if}\;t \leq -5.2 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+149}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) t)))
   (if (<= t -5.2e+15) t_1 (if (<= t 3e+149) (* (/ x z) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * t;
	double tmp;
	if (t <= -5.2e+15) {
		tmp = t_1;
	} else if (t <= 3e+149) {
		tmp = (x / z) * y;
	} 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 / z) * t
    if (t <= (-5.2d+15)) then
        tmp = t_1
    else if (t <= 3d+149) then
        tmp = (x / z) * y
    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 / z) * t;
	double tmp;
	if (t <= -5.2e+15) {
		tmp = t_1;
	} else if (t <= 3e+149) {
		tmp = (x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / z) * t
	tmp = 0
	if t <= -5.2e+15:
		tmp = t_1
	elif t <= 3e+149:
		tmp = (x / z) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * t)
	tmp = 0.0
	if (t <= -5.2e+15)
		tmp = t_1;
	elseif (t <= 3e+149)
		tmp = Float64(Float64(x / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / z) * t;
	tmp = 0.0;
	if (t <= -5.2e+15)
		tmp = t_1;
	elseif (t <= 3e+149)
		tmp = (x / z) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -5.2e+15], t$95$1, If[LessEqual[t, 3e+149], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.2e15 or 3.00000000000000003e149 < t

   1. Initial program 96.1%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   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 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 46.3%

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

   if -5.2e15 < t < 3.00000000000000003e149

   1. Initial program 92.9%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 80.6

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

   5. Applied rewrites 80.6%

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

   7. Applied rewrites 80.7%

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

Alternative 11: 43.2% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x}{z} \cdot t\\ \mathbf{if}\;z \leq -130000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ x z) t)))
   (if (<= z -130000.0) t_1 (if (<= z 2.45e-18) (* (- t) (fma z x x)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / z) * t;
	double tmp;
	if (z <= -130000.0) {
		tmp = t_1;
	} else if (z <= 2.45e-18) {
		tmp = -t * fma(z, x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / z) * t)
	tmp = 0.0
	if (z <= -130000.0)
		tmp = t_1;
	elseif (z <= 2.45e-18)
		tmp = Float64(Float64(-t) * fma(z, x, x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -130000.0], t$95$1, If[LessEqual[z, 2.45e-18], N[((-t) * N[(z * x + x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3e5 or 2.4500000000000001e-18 < z

   1. Initial program 96.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   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 48.7

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

   5. Applied rewrites 48.7%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.3%

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

   if -1.3e5 < z < 2.4500000000000001e-18

   1. Initial program 90.8%

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

   3. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{t \cdot x}{1 - z}} \]
   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 34.9

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

   5. Applied rewrites 34.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 34.1%

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

4. Final simplification 41.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -130000:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \mathbf{elif}\;z \leq 2.45 \cdot 10^{-18}:\\ \;\;\;\;\left(-t\right) \cdot \mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 23.7% 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 93.9%

   \[x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right) \]
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. sub-neg N/A

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

   11. lower--.f64 44.5

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

5. Applied rewrites 44.5%

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

6. Taylor expanded in z 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. Final simplification 21.6%

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

Developer Target 1: 95.1% 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 2024296 
(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)))))