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

Percentage Accurate: 94.5% → 96.6%

Time: 9.0s

Alternatives: 11

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.6% accurate, 0.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 47.0%

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

   3. Taylor expanded in y around inf 99.8%

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

      1. *-commutative 99.8%

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

      2. associate-/l* 99.7%

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

   5. Applied egg-rr 99.7%

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

      1. clear-num 99.6%

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

      2. un-div-inv 99.8%

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

   7. Applied egg-rr 99.8%

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

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

   1. Initial program 99.0%

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

4. Final simplification 99.1%

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

Alternative 2: 93.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 0.006:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{z}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.0)
   (* x (/ (+ y t) z))
   (if (<= z 0.006) (* x (- (/ y z) t)) (* x (+ (/ y z) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.0) {
		tmp = x * ((y + t) / z);
	} else if (z <= 0.006) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * ((y / z) + (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.0d0)) then
        tmp = x * ((y + t) / z)
    else if (z <= 0.006d0) then
        tmp = x * ((y / z) - t)
    else
        tmp = x * ((y / z) + (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.0) {
		tmp = x * ((y + t) / z);
	} else if (z <= 0.006) {
		tmp = x * ((y / z) - t);
	} else {
		tmp = x * ((y / z) + (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.0:
		tmp = x * ((y + t) / z)
	elif z <= 0.006:
		tmp = x * ((y / z) - t)
	else:
		tmp = x * ((y / z) + (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.0)
		tmp = Float64(x * Float64(Float64(y + t) / z));
	elseif (z <= 0.006)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	else
		tmp = Float64(x * Float64(Float64(y / z) + Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.0)
		tmp = x * ((y + t) / z);
	elseif (z <= 0.006)
		tmp = x * ((y / z) - t);
	else
		tmp = x * ((y / z) + (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.0], N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.006], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / z), $MachinePrecision] + N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1

   1. Initial program 98.5%

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

   3. Taylor expanded in z around inf 87.8%

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

      1. associate-/l* 97.8%

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

      2. cancel-sign-sub-inv 97.8%

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

      3. metadata-eval 97.8%

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

      4. *-lft-identity 97.8%

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

      5. +-commutative 97.8%

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

   5. Simplified 97.8%

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

   if -1 < z < 0.0060000000000000001

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0 92.0%

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

   if 0.0060000000000000001 < z

   1. Initial program 99.7%

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

   3. Taylor expanded in z around inf 99.6%

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

      1. associate-*r/ 99.6%

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

      2. neg-mul-1 99.6%

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

   5. Simplified 99.6%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq 0.006:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} + \frac{t}{z}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.006\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 0.006)))
   (* x (/ (+ y t) z))
   (* x (- (/ y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.006)) {
		tmp = x * ((y + t) / z);
	} else {
		tmp = x * ((y / z) - t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 0.006d0))) then
        tmp = x * ((y + t) / z)
    else
        tmp = x * ((y / z) - t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.0) || !(z <= 0.006)) {
		tmp = x * ((y + t) / z);
	} else {
		tmp = x * ((y / z) - t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.0) or not (z <= 0.006):
		tmp = x * ((y + t) / z)
	else:
		tmp = x * ((y / z) - t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 0.006))
		tmp = Float64(x * Float64(Float64(y + t) / z));
	else
		tmp = Float64(x * Float64(Float64(y / z) - t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 0.006)))
		tmp = x * ((y + t) / z);
	else
		tmp = x * ((y / z) - t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 0.006]], $MachinePrecision]], N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 0.0060000000000000001 < z

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf 86.1%

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

      1. associate-/l* 98.6%

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

      2. cancel-sign-sub-inv 98.6%

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

      3. metadata-eval 98.6%

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

      4. *-lft-identity 98.6%

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

      5. +-commutative 98.6%

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

   5. Simplified 98.6%

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

   if -1 < z < 0.0060000000000000001

   1. Initial program 92.0%

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

   3. Taylor expanded in z around 0 92.0%

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

4. Final simplification 95.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 0.006\right):\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 72.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 0.112:\\ \;\;\;\;t \cdot \frac{x}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.9e-123)
   (* y (/ x z))
   (if (<= y 0.112) (* t (/ x (+ z -1.0))) (/ y (/ z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.9e-123) {
		tmp = y * (x / z);
	} else if (y <= 0.112) {
		tmp = t * (x / (z + -1.0));
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.9d-123)) then
        tmp = y * (x / z)
    else if (y <= 0.112d0) then
        tmp = t * (x / (z + (-1.0d0)))
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.9e-123) {
		tmp = y * (x / z);
	} else if (y <= 0.112) {
		tmp = t * (x / (z + -1.0));
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.9e-123:
		tmp = y * (x / z)
	elif y <= 0.112:
		tmp = t * (x / (z + -1.0))
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.9e-123)
		tmp = Float64(y * Float64(x / z));
	elseif (y <= 0.112)
		tmp = Float64(t * Float64(x / Float64(z + -1.0)));
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.9e-123)
		tmp = y * (x / z);
	elseif (y <= 0.112)
		tmp = t * (x / (z + -1.0));
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.9e-123], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.112], N[(t * N[(x / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.89999999999999998e-123

   1. Initial program 93.4%

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

   3. Taylor expanded in y around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-/l* 79.3%

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

   5. Applied egg-rr 79.3%

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

   if -1.89999999999999998e-123 < y < 0.112000000000000002

   1. Initial program 98.8%

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

   3. Taylor expanded in y around 0 79.8%

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

      1. mul-1-neg 79.8%

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

      2. associate-/l* 77.1%

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

      3. distribute-rgt-neg-in 77.1%

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

      4. distribute-neg-frac2 77.1%

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

      5. neg-sub0 77.1%

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

      6. associate--r- 77.1%

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

      7. metadata-eval 77.1%

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

   5. Simplified 77.1%

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

   if 0.112000000000000002 < y

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-/l* 80.6%

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

   5. Applied egg-rr 80.6%

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

      1. clear-num 80.5%

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

      2. un-div-inv 80.8%

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

   7. Applied egg-rr 80.8%

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

4. Final simplification 78.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 0.112:\\ \;\;\;\;t \cdot \frac{x}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 64.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-123} \lor \neg \left(y \leq 0.112\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.8e-123) (not (<= y 0.112))) (* y (/ x z)) (* x (/ t z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.8e-123) || !(y <= 0.112)) {
		tmp = y * (x / z);
	} else {
		tmp = x * (t / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((y <= (-1.8d-123)) .or. (.not. (y <= 0.112d0))) then
        tmp = y * (x / z)
    else
        tmp = x * (t / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.8e-123) || !(y <= 0.112)) {
		tmp = y * (x / z);
	} else {
		tmp = x * (t / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -1.8e-123) or not (y <= 0.112):
		tmp = y * (x / z)
	else:
		tmp = x * (t / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.8e-123) || !(y <= 0.112))
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(x * Float64(t / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((y <= -1.8e-123) || ~((y <= 0.112)))
		tmp = y * (x / z);
	else
		tmp = x * (t / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.8e-123], N[Not[LessEqual[y, 0.112]], $MachinePrecision]], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.7999999999999998e-123 or 0.112000000000000002 < y

   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 inf 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-/l* 79.9%

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

   5. Applied egg-rr 79.9%

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

   if -1.7999999999999998e-123 < y < 0.112000000000000002

   1. Initial program 98.8%

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

   3. Taylor expanded in z around inf 61.1%

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

      1. associate-/l* 63.9%

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

      2. cancel-sign-sub-inv 63.9%

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

      3. metadata-eval 63.9%

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

      4. *-lft-identity 63.9%

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

      5. +-commutative 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in t around inf 58.2%

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

4. Final simplification 71.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.8 \cdot 10^{-123} \lor \neg \left(y \leq 0.112\right):\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 65.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{+246} \lor \neg \left(t \leq 7 \cdot 10^{+241}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.15e+246) (not (<= t 7e+241))) (* x (/ t z)) (* (/ y z) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.15e+246) || !(t <= 7e+241)) {
		tmp = x * (t / z);
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((t <= (-2.15d+246)) .or. (.not. (t <= 7d+241))) then
        tmp = x * (t / z)
    else
        tmp = (y / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.15e+246) || !(t <= 7e+241)) {
		tmp = x * (t / z);
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.15e+246) or not (t <= 7e+241):
		tmp = x * (t / z)
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.15e+246) || !(t <= 7e+241))
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.15e+246) || ~((t <= 7e+241)))
		tmp = x * (t / z);
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.15e+246], N[Not[LessEqual[t, 7e+241]], $MachinePrecision]], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.15000000000000014e246 or 7e241 < t

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf 54.1%

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

      1. associate-/l* 65.8%

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

      2. cancel-sign-sub-inv 65.8%

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

      3. metadata-eval 65.8%

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

      4. *-lft-identity 65.8%

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

      5. +-commutative 65.8%

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

   5. Simplified 65.8%

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

   6. Taylor expanded in t around inf 64.3%

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

   if -2.15000000000000014e246 < t < 7e241

   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 inf 68.2%

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

      1. associate-*r/ 71.9%

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

   5. Simplified 71.9%

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

4. Final simplification 70.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.15 \cdot 10^{+246} \lor \neg \left(t \leq 7 \cdot 10^{+241}\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 45.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 850\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 850.0))) (* x (/ t z)) (* x (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.0) || !(z <= 850.0)) {
		tmp = x * (t / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 850.0d0))) then
        tmp = x * (t / z)
    else
        tmp = x * -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.0) || !(z <= 850.0)) {
		tmp = x * (t / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.0) or not (z <= 850.0):
		tmp = x * (t / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 850.0))
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 850.0)))
		tmp = x * (t / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 850.0]], $MachinePrecision]], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 850 < z

   1. Initial program 99.0%

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

   3. Taylor expanded in z around inf 85.9%

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

      1. associate-/l* 98.6%

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

      2. cancel-sign-sub-inv 98.6%

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

      3. metadata-eval 98.6%

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

      4. *-lft-identity 98.6%

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

      5. +-commutative 98.6%

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

   5. Simplified 98.6%

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

   6. Taylor expanded in t around inf 58.9%

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

   if -1 < z < 850

   1. Initial program 92.2%

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

   3. Taylor expanded in y around 0 34.5%

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

      1. mul-1-neg 34.5%

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

      2. associate-/l* 34.5%

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

      3. distribute-rgt-neg-in 34.5%

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

      4. distribute-neg-frac2 34.5%

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

      5. neg-sub0 34.5%

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

      6. associate--r- 34.5%

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

      7. metadata-eval 34.5%

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

   5. Simplified 34.5%

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

   6. Taylor expanded in z around 0 34.6%

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

      1. neg-mul-1 34.6%

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

   8. Simplified 34.6%

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

4. Final simplification 47.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 850\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 42.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 850\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -1.0) (not (<= z 850.0))) (* t (/ x z)) (* x (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.0) || !(z <= 850.0)) {
		tmp = t * (x / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-1.0d0)) .or. (.not. (z <= 850.0d0))) then
        tmp = t * (x / z)
    else
        tmp = x * -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -1.0) || !(z <= 850.0)) {
		tmp = t * (x / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -1.0) or not (z <= 850.0):
		tmp = t * (x / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -1.0) || !(z <= 850.0))
		tmp = Float64(t * Float64(x / z));
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -1.0) || ~((z <= 850.0)))
		tmp = t * (x / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -1.0], N[Not[LessEqual[z, 850.0]], $MachinePrecision]], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1 or 850 < z

   1. Initial program 99.0%

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

   3. Taylor expanded in y around 0 54.3%

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

      1. mul-1-neg 54.3%

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

      2. associate-/l* 53.1%

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

      3. distribute-rgt-neg-in 53.1%

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

      4. distribute-neg-frac2 53.1%

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

      5. neg-sub0 53.1%

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

      6. associate--r- 53.1%

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

      7. metadata-eval 53.1%

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

   5. Simplified 53.1%

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

   6. Taylor expanded in z around inf 52.6%

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

   if -1 < z < 850

   1. Initial program 92.2%

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

   3. Taylor expanded in y around 0 34.5%

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

      1. mul-1-neg 34.5%

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

      2. associate-/l* 34.5%

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

      3. distribute-rgt-neg-in 34.5%

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

      4. distribute-neg-frac2 34.5%

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

      5. neg-sub0 34.5%

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

      6. associate--r- 34.5%

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

      7. metadata-eval 34.5%

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

   5. Simplified 34.5%

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

   6. Taylor expanded in z around 0 34.6%

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

      1. neg-mul-1 34.6%

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

   8. Simplified 34.6%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \lor \neg \left(z \leq 850\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.9 \cdot 10^{-123}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 0.112:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.9e-123)
   (* y (/ x z))
   (if (<= y 0.112) (* x (/ t z)) (/ y (/ z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.9e-123) {
		tmp = y * (x / z);
	} else if (y <= 0.112) {
		tmp = x * (t / z);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.9d-123)) then
        tmp = y * (x / z)
    else if (y <= 0.112d0) then
        tmp = x * (t / z)
    else
        tmp = y / (z / x)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.9e-123) {
		tmp = y * (x / z);
	} else if (y <= 0.112) {
		tmp = x * (t / z);
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.9e-123:
		tmp = y * (x / z)
	elif y <= 0.112:
		tmp = x * (t / z)
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.9e-123)
		tmp = Float64(y * Float64(x / z));
	elseif (y <= 0.112)
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.9e-123)
		tmp = y * (x / z);
	elseif (y <= 0.112)
		tmp = x * (t / z);
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.9e-123], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 0.112], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.89999999999999998e-123

   1. Initial program 93.4%

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

   3. Taylor expanded in y around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-/l* 79.3%

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

   5. Applied egg-rr 79.3%

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

   if -1.89999999999999998e-123 < y < 0.112000000000000002

   1. Initial program 98.8%

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

   3. Taylor expanded in z around inf 61.1%

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

      1. associate-/l* 63.9%

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

      2. cancel-sign-sub-inv 63.9%

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

      3. metadata-eval 63.9%

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

      4. *-lft-identity 63.9%

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

      5. +-commutative 63.9%

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

   5. Simplified 63.9%

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

   6. Taylor expanded in t around inf 58.2%

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

   if 0.112000000000000002 < y

   1. Initial program 94.5%

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

   3. Taylor expanded in y around inf 78.1%

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

      1. *-commutative 78.1%

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

      2. associate-/l* 80.6%

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

   5. Applied egg-rr 80.6%

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

      1. clear-num 80.5%

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

      2. un-div-inv 80.8%

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

   7. Applied egg-rr 80.8%

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

Alternative 10: 24.5% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 850:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z 850.0) (* x (- t)) (* t x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 850.0) {
		tmp = x * -t;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= 850.0d0) then
        tmp = x * -t
    else
        tmp = t * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= 850.0) {
		tmp = x * -t;
	} else {
		tmp = t * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= 850.0:
		tmp = x * -t
	else:
		tmp = t * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= 850.0)
		tmp = Float64(x * Float64(-t));
	else
		tmp = Float64(t * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= 850.0)
		tmp = x * -t;
	else
		tmp = t * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, 850.0], N[(x * (-t)), $MachinePrecision], N[(t * x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 850

   1. Initial program 94.6%

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

   3. Taylor expanded in y around 0 43.4%

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

      1. mul-1-neg 43.4%

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

      2. associate-/l* 42.4%

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

      3. distribute-rgt-neg-in 42.4%

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

      4. distribute-neg-frac2 42.4%

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

      5. neg-sub0 42.4%

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

      6. associate--r- 42.4%

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

      7. metadata-eval 42.4%

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

   5. Simplified 42.4%

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

   6. Taylor expanded in z around 0 27.7%

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

      1. neg-mul-1 27.7%

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

   8. Simplified 27.7%

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

   if 850 < z

   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 49.8%

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

      1. mul-1-neg 49.8%

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

      2. associate-/l* 50.3%

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

      3. distribute-rgt-neg-in 50.3%

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

      4. distribute-neg-frac2 50.3%

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

      5. neg-sub0 50.3%

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

      6. associate--r- 50.3%

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

      7. metadata-eval 50.3%

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

   5. Simplified 50.3%

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

   6. Taylor expanded in z around 0 6.9%

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

      1. neg-mul-1 6.9%

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

   8. Simplified 6.9%

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

      1. neg-sub0 6.9%

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

      2. sub-neg 6.9%

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

      3. add-sqr-sqrt 3.3%

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

      4. sqrt-unprod 17.0%

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

      5. sqr-neg 17.0%

         \[\leadsto t \cdot \left(0 + \sqrt{\color{blue}{x \cdot x}}\right) \]

      6. sqrt-unprod 7.3%

         \[\leadsto t \cdot \left(0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) \]

      7. add-sqr-sqrt 14.5%

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

   10. Applied egg-rr 14.5%

       \[\leadsto t \cdot \color{blue}{\left(0 + x\right)} \]
   11. Step-by-step derivation

       1. +-lft-identity 14.5%

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

   12. Simplified 14.5%

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

4. Final simplification 24.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 850:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 9.7% accurate, 3.7× speedup

Math

\[\begin{array}{l} \\ t \cdot x \end{array} \]

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(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 95.8%

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

3. Taylor expanded in y around 0 44.9%

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

   1. mul-1-neg 44.9%

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

   2. associate-/l* 44.2%

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

   3. distribute-rgt-neg-in 44.2%

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

   4. distribute-neg-frac2 44.2%

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

   5. neg-sub0 44.2%

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

   6. associate--r- 44.2%

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

   7. metadata-eval 44.2%

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

5. Simplified 44.2%

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

6. Taylor expanded in z around 0 22.9%

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

   1. neg-mul-1 22.9%

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

8. Simplified 22.9%

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

   1. neg-sub0 22.9%

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

   2. sub-neg 22.9%

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

   3. add-sqr-sqrt 11.0%

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

   4. sqrt-unprod 22.3%

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

   5. sqr-neg 22.3%

      \[\leadsto t \cdot \left(0 + \sqrt{\color{blue}{x \cdot x}}\right) \]

   6. sqrt-unprod 5.8%

      \[\leadsto t \cdot \left(0 + \color{blue}{\sqrt{x} \cdot \sqrt{x}}\right) \]

   7. add-sqr-sqrt 12.3%

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

10. Applied egg-rr 12.3%

    \[\leadsto t \cdot \color{blue}{\left(0 + x\right)} \]
11. Step-by-step derivation

    1. +-lft-identity 12.3%

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

12. Simplified 12.3%

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

Developer Target 1: 94.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t \cdot \frac{1}{1 - z}\right)\\ t_2 := x \cdot \left(\frac{y}{z} - \frac{t}{1 - z}\right)\\ \mathbf{if}\;t\_2 < -7.623226303312042 \cdot 10^{-196}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 < 1.4133944927702302 \cdot 10^{-211}:\\ \;\;\;\;\frac{y \cdot x}{z} + \left(-\frac{t \cdot x}{1 - z}\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))))
        (t_2 (* x (- (/ y z) (/ t (- 1.0 z))))))
   (if (< t_2 -7.623226303312042e-196)
     t_1
     (if (< t_2 1.4133944927702302e-211)
       (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z))))
       t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * ((y / z) - (t * (1.0d0 / (1.0d0 - z))))
    t_2 = x * ((y / z) - (t / (1.0d0 - z)))
    if (t_2 < (-7.623226303312042d-196)) then
        tmp = t_1
    else if (t_2 < 1.4133944927702302d-211) then
        tmp = ((y * x) / z) + -((t * x) / (1.0d0 - z))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	double t_2 = x * ((y / z) - (t / (1.0 - z)));
	double tmp;
	if (t_2 < -7.623226303312042e-196) {
		tmp = t_1;
	} else if (t_2 < 1.4133944927702302e-211) {
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))))
	t_2 = x * ((y / z) - (t / (1.0 - z)))
	tmp = 0
	if t_2 < -7.623226303312042e-196:
		tmp = t_1
	elif t_2 < 1.4133944927702302e-211:
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y / z) - Float64(t * Float64(1.0 / Float64(1.0 - z)))))
	t_2 = Float64(x * Float64(Float64(y / z) - Float64(t / Float64(1.0 - z))))
	tmp = 0.0
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = Float64(Float64(Float64(y * x) / z) + Float64(-Float64(Float64(t * x) / Float64(1.0 - z))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) - (t * (1.0 / (1.0 - z))));
	t_2 = x * ((y / z) - (t / (1.0 - z)));
	tmp = 0.0;
	if (t_2 < -7.623226303312042e-196)
		tmp = t_1;
	elseif (t_2 < 1.4133944927702302e-211)
		tmp = ((y * x) / z) + -((t * x) / (1.0 - z));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t * N[(1.0 / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[t$95$2, -7.623226303312042e-196], t$95$1, If[Less[t$95$2, 1.4133944927702302e-211], N[(N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision] + (-N[(N[(t * x), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision])), $MachinePrecision], t$95$1]]]]

Reproduce

herbie shell --seed 2024152 
(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)))))