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

Percentage Accurate: 94.3% → 94.3%

Time: 8.9s

Alternatives: 9

Speedup: 1.0×

• 20.3% 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: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Final simplification 96.9%

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

Alternative 2: 64.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ t_2 := t \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -3.15 \cdot 10^{-128}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-229}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-127}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-121}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ y z))) (t_2 (* t (/ x z))))
   (if (<= y -3.15e-128)
     t_1
     (if (<= y 8.5e-229)
       t_2
       (if (<= y 7.5e-127) (* x (- t)) (if (<= y 3.7e-121) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y / z);
	double t_2 = t * (x / z);
	double tmp;
	if (y <= -3.15e-128) {
		tmp = t_1;
	} else if (y <= 8.5e-229) {
		tmp = t_2;
	} else if (y <= 7.5e-127) {
		tmp = x * -t;
	} else if (y <= 3.7e-121) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * (y / z)
    t_2 = t * (x / z)
    if (y <= (-3.15d-128)) then
        tmp = t_1
    else if (y <= 8.5d-229) then
        tmp = t_2
    else if (y <= 7.5d-127) then
        tmp = x * -t
    else if (y <= 3.7d-121) then
        tmp = t_2
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (y / z);
	double t_2 = t * (x / z);
	double tmp;
	if (y <= -3.15e-128) {
		tmp = t_1;
	} else if (y <= 8.5e-229) {
		tmp = t_2;
	} else if (y <= 7.5e-127) {
		tmp = x * -t;
	} else if (y <= 3.7e-121) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (y / z)
	t_2 = t * (x / z)
	tmp = 0
	if y <= -3.15e-128:
		tmp = t_1
	elif y <= 8.5e-229:
		tmp = t_2
	elif y <= 7.5e-127:
		tmp = x * -t
	elif y <= 3.7e-121:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y / z))
	t_2 = Float64(t * Float64(x / z))
	tmp = 0.0
	if (y <= -3.15e-128)
		tmp = t_1;
	elseif (y <= 8.5e-229)
		tmp = t_2;
	elseif (y <= 7.5e-127)
		tmp = Float64(x * Float64(-t));
	elseif (y <= 3.7e-121)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y / z);
	t_2 = t * (x / z);
	tmp = 0.0;
	if (y <= -3.15e-128)
		tmp = t_1;
	elseif (y <= 8.5e-229)
		tmp = t_2;
	elseif (y <= 7.5e-127)
		tmp = x * -t;
	elseif (y <= 3.7e-121)
		tmp = t_2;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.15e-128], t$95$1, If[LessEqual[y, 8.5e-229], t$95$2, If[LessEqual[y, 7.5e-127], N[(x * (-t)), $MachinePrecision], If[LessEqual[y, 3.7e-121], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.1500000000000002e-128 or 3.7000000000000002e-121 < y

   1. Initial program 97.7%

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

   3. Taylor expanded in y around inf 77.9%

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

      1. associate-*r/ 80.6%

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

   5. Simplified 80.6%

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

   if -3.1500000000000002e-128 < y < 8.49999999999999977e-229 or 7.5000000000000004e-127 < y < 3.7000000000000002e-121

   1. Initial program 94.3%

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

   3. Taylor expanded in y around 0 87.5%

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

      1. mul-1-neg 87.5%

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

      2. distribute-neg-frac2 87.5%

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

      3. neg-sub0 87.5%

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

      4. associate--r- 87.5%

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

      5. metadata-eval 87.5%

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

   5. Simplified 87.5%

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

   6. Taylor expanded in z around inf 62.5%

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

      1. associate-/l* 68.0%

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

   8. Simplified 68.0%

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

   if 8.49999999999999977e-229 < y < 7.5000000000000004e-127

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.3%

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

      1. mul-1-neg 65.3%

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

      2. distribute-neg-frac2 65.3%

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

      3. neg-sub0 65.3%

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

      4. associate--r- 65.3%

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

      5. metadata-eval 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in z around 0 56.7%

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

      1. *-commutative 56.7%

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

      2. neg-mul-1 56.7%

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

      3. distribute-lft-neg-in 56.7%

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

   8. Simplified 56.7%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.15 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 8.5 \cdot 10^{-229}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{-127}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{-121}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 64.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{x}{z}\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{-129}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-230}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-120}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* t (/ x z))))
   (if (<= y -3.5e-129)
     (* x (/ y z))
     (if (<= y 1.45e-230)
       t_1
       (if (<= y 2.6e-130)
         (* x (- t))
         (if (<= y 8.4e-120) t_1 (* y (/ x z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = t * (x / z);
	double tmp;
	if (y <= -3.5e-129) {
		tmp = x * (y / z);
	} else if (y <= 1.45e-230) {
		tmp = t_1;
	} else if (y <= 2.6e-130) {
		tmp = x * -t;
	} else if (y <= 8.4e-120) {
		tmp = t_1;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t * (x / z)
    if (y <= (-3.5d-129)) then
        tmp = x * (y / z)
    else if (y <= 1.45d-230) then
        tmp = t_1
    else if (y <= 2.6d-130) then
        tmp = x * -t
    else if (y <= 8.4d-120) then
        tmp = t_1
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = t * (x / z);
	double tmp;
	if (y <= -3.5e-129) {
		tmp = x * (y / z);
	} else if (y <= 1.45e-230) {
		tmp = t_1;
	} else if (y <= 2.6e-130) {
		tmp = x * -t;
	} else if (y <= 8.4e-120) {
		tmp = t_1;
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = t * (x / z)
	tmp = 0
	if y <= -3.5e-129:
		tmp = x * (y / z)
	elif y <= 1.45e-230:
		tmp = t_1
	elif y <= 2.6e-130:
		tmp = x * -t
	elif y <= 8.4e-120:
		tmp = t_1
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(t * Float64(x / z))
	tmp = 0.0
	if (y <= -3.5e-129)
		tmp = Float64(x * Float64(y / z));
	elseif (y <= 1.45e-230)
		tmp = t_1;
	elseif (y <= 2.6e-130)
		tmp = Float64(x * Float64(-t));
	elseif (y <= 8.4e-120)
		tmp = t_1;
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = t * (x / z);
	tmp = 0.0;
	if (y <= -3.5e-129)
		tmp = x * (y / z);
	elseif (y <= 1.45e-230)
		tmp = t_1;
	elseif (y <= 2.6e-130)
		tmp = x * -t;
	elseif (y <= 8.4e-120)
		tmp = t_1;
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3.5e-129], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.45e-230], t$95$1, If[LessEqual[y, 2.6e-130], N[(x * (-t)), $MachinePrecision], If[LessEqual[y, 8.4e-120], t$95$1, N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.4999999999999997e-129

   1. Initial program 98.9%

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

   3. Taylor expanded in y around inf 81.0%

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

      1. associate-*r/ 86.1%

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

   5. Simplified 86.1%

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

   if -3.4999999999999997e-129 < y < 1.45000000000000003e-230 or 2.6000000000000001e-130 < y < 8.4000000000000002e-120

   1. Initial program 94.3%

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

   3. Taylor expanded in y around 0 87.5%

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

      1. mul-1-neg 87.5%

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

      2. distribute-neg-frac2 87.5%

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

      3. neg-sub0 87.5%

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

      4. associate--r- 87.5%

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

      5. metadata-eval 87.5%

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

   5. Simplified 87.5%

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

   6. Taylor expanded in z around inf 62.5%

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

      1. associate-/l* 68.0%

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

   8. Simplified 68.0%

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

   if 1.45000000000000003e-230 < y < 2.6000000000000001e-130

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0 65.3%

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

      1. mul-1-neg 65.3%

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

      2. distribute-neg-frac2 65.3%

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

      3. neg-sub0 65.3%

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

      4. associate--r- 65.3%

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

      5. metadata-eval 65.3%

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

   5. Simplified 65.3%

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

   6. Taylor expanded in z around 0 56.7%

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

      1. *-commutative 56.7%

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

      2. neg-mul-1 56.7%

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

      3. distribute-lft-neg-in 56.7%

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

   8. Simplified 56.7%

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

   if 8.4000000000000002e-120 < y

   1. Initial program 96.4%

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

   3. Taylor expanded in y around inf 74.3%

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

      1. clear-num 73.5%

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

      2. associate-/r/ 74.2%

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

      3. associate-*r* 75.3%

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

      4. associate-*l/ 75.4%

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

      5. *-un-lft-identity 75.4%

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

   5. Applied egg-rr 75.4%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{-129}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.45 \cdot 10^{-230}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-130}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \mathbf{elif}\;y \leq 8.4 \cdot 10^{-120}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.6% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if z < -1 or 1 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf 87.9%

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

      1. sub-neg 87.9%

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

      2. remove-double-neg 87.9%

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

      3. neg-mul-1 87.9%

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

      4. distribute-neg-in 87.9%

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

      5. neg-mul-1 87.9%

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

      6. sub-neg 87.9%

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

      7. distribute-rgt-neg-in 87.9%

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

      8. distribute-neg-frac 87.9%

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

      9. associate-/l* 96.1%

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

      10. distribute-rgt-neg-in 96.1%

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

      11. distribute-neg-frac 96.1%

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

      12. sub-neg 96.1%

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

      13. neg-mul-1 96.1%

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

      14. distribute-neg-in 96.1%

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

      15. remove-double-neg 96.1%

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

      16. remove-double-neg 96.1%

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

      17. +-commutative 96.1%

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

   5. Simplified 96.1%

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

   if -1 < z < 1

   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 0 96.2%

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

      1. mul-1-neg 96.2%

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

      2. unsub-neg 96.2%

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

      3. div-sub 96.2%

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

      4. associate-/l* 96.2%

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

      5. *-inverses 96.2%

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

      6. *-rgt-identity 96.2%

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

   5. Simplified 96.2%

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

4. Final simplification 96.1%

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

Alternative 5: 73.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-111}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-74}:\\ \;\;\;\;t \cdot \frac{x}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.2e-111)
   (* x (/ y z))
   (if (<= y 6.9e-74) (* t (/ x (+ z -1.0))) (* y (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.2e-111) {
		tmp = x * (y / z);
	} else if (y <= 6.9e-74) {
		tmp = t * (x / (z + -1.0));
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-7.2d-111)) then
        tmp = x * (y / z)
    else if (y <= 6.9d-74) then
        tmp = t * (x / (z + (-1.0d0)))
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.2e-111) {
		tmp = x * (y / z);
	} else if (y <= 6.9e-74) {
		tmp = t * (x / (z + -1.0));
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -7.2e-111:
		tmp = x * (y / z)
	elif y <= 6.9e-74:
		tmp = t * (x / (z + -1.0))
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.2e-111)
		tmp = Float64(x * Float64(y / z));
	elseif (y <= 6.9e-74)
		tmp = Float64(t * Float64(x / Float64(z + -1.0)));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -7.2e-111)
		tmp = x * (y / z);
	elseif (y <= 6.9e-74)
		tmp = t * (x / (z + -1.0));
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.2e-111], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 6.9e-74], N[(t * N[(x / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.20000000000000019e-111

   1. Initial program 98.9%

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

   3. Taylor expanded in y around inf 81.0%

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

      1. associate-*r/ 86.1%

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

   5. Simplified 86.1%

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

   if -7.20000000000000019e-111 < y < 6.89999999999999981e-74

   1. Initial program 95.9%

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

   3. Taylor expanded in y around 0 78.6%

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

      1. mul-1-neg 78.6%

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

      2. associate-/l* 81.7%

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

      3. distribute-rgt-neg-in 81.7%

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

      4. distribute-neg-frac2 81.7%

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

      5. neg-sub0 81.7%

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

      6. associate--r- 81.7%

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

      7. metadata-eval 81.7%

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

   5. Simplified 81.7%

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

   if 6.89999999999999981e-74 < y

   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 inf 78.3%

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

      1. clear-num 78.1%

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

      2. associate-/r/ 78.1%

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

      3. associate-*r* 79.6%

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

      4. associate-*l/ 79.7%

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

      5. *-un-lft-identity 79.7%

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

   5. Applied egg-rr 79.7%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{-111}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{-74}:\\ \;\;\;\;t \cdot \frac{x}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 74.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-69}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.12e-100)
   (* x (/ y z))
   (if (<= y 1.65e-69) (* x (/ t (+ z -1.0))) (* y (/ x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.12e-100) {
		tmp = x * (y / z);
	} else if (y <= 1.65e-69) {
		tmp = x * (t / (z + -1.0));
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.12d-100)) then
        tmp = x * (y / z)
    else if (y <= 1.65d-69) then
        tmp = x * (t / (z + (-1.0d0)))
    else
        tmp = y * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.12e-100) {
		tmp = x * (y / z);
	} else if (y <= 1.65e-69) {
		tmp = x * (t / (z + -1.0));
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.12e-100:
		tmp = x * (y / z)
	elif y <= 1.65e-69:
		tmp = x * (t / (z + -1.0))
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.12e-100)
		tmp = Float64(x * Float64(y / z));
	elseif (y <= 1.65e-69)
		tmp = Float64(x * Float64(t / Float64(z + -1.0)));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.12e-100)
		tmp = x * (y / z);
	elseif (y <= 1.65e-69)
		tmp = x * (t / (z + -1.0));
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.12e-100], N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.65e-69], N[(x * N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.11999999999999996e-100

   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 inf 82.3%

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

      1. associate-*r/ 87.6%

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

   5. Simplified 87.6%

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

   if -1.11999999999999996e-100 < y < 1.65e-69

   1. Initial program 96.0%

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

   3. Taylor expanded in y around 0 81.3%

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

      1. mul-1-neg 81.3%

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

      2. distribute-neg-frac2 81.3%

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

      3. neg-sub0 81.3%

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

      4. associate--r- 81.3%

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

      5. metadata-eval 81.3%

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

   5. Simplified 81.3%

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

   if 1.65e-69 < y

   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 inf 78.3%

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

      1. clear-num 78.1%

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

      2. associate-/r/ 78.1%

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

      3. associate-*r* 79.6%

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

      4. associate-*l/ 79.7%

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

      5. *-un-lft-identity 79.7%

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

   5. Applied egg-rr 79.7%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.12 \cdot 10^{-100}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;y \leq 1.65 \cdot 10^{-69}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 42.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-18} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.3e-18) (not (<= z 1.0))) (* t (/ x z)) (* x (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.3e-18) || !(z <= 1.0)) {
		tmp = t * (x / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((z <= (-2.3d-18)) .or. (.not. (z <= 1.0d0))) then
        tmp = t * (x / z)
    else
        tmp = x * -t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.3e-18) || !(z <= 1.0)) {
		tmp = t * (x / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.3e-18) or not (z <= 1.0):
		tmp = t * (x / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.3e-18) || !(z <= 1.0))
		tmp = Float64(t * Float64(x / z));
	else
		tmp = Float64(x * Float64(-t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -2.3e-18) || ~((z <= 1.0)))
		tmp = t * (x / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.3e-18], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3000000000000001e-18 or 1 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0 56.7%

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

      1. mul-1-neg 56.7%

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

      2. distribute-neg-frac2 56.7%

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

      3. neg-sub0 56.7%

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

      4. associate--r- 56.7%

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

      5. metadata-eval 56.7%

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

   5. Simplified 56.7%

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

   6. Taylor expanded in z around inf 52.6%

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

      1. associate-/l* 52.1%

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

   8. Simplified 52.1%

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

   if -2.3000000000000001e-18 < z < 1

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0 35.5%

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

      1. mul-1-neg 35.5%

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

      2. distribute-neg-frac2 35.5%

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

      3. neg-sub0 35.5%

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

      4. associate--r- 35.5%

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

      5. metadata-eval 35.5%

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

   5. Simplified 35.5%

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

   6. Taylor expanded in z around 0 34.8%

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

      1. *-commutative 34.8%

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

      2. neg-mul-1 34.8%

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

      3. distribute-lft-neg-in 34.8%

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

   8. Simplified 34.8%

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

4. Final simplification 44.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-18} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 44.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-18} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -2.3e-18) (not (<= z 1.0))) (* x (/ t z)) (* x (- t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -2.3e-18) || !(z <= 1.0)) {
		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 <= (-2.3d-18)) .or. (.not. (z <= 1.0d0))) 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 <= -2.3e-18) || !(z <= 1.0)) {
		tmp = x * (t / z);
	} else {
		tmp = x * -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -2.3e-18) or not (z <= 1.0):
		tmp = x * (t / z)
	else:
		tmp = x * -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -2.3e-18) || !(z <= 1.0))
		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 <= -2.3e-18) || ~((z <= 1.0)))
		tmp = x * (t / z);
	else
		tmp = x * -t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -2.3e-18], N[Not[LessEqual[z, 1.0]], $MachinePrecision]], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(x * (-t)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.3000000000000001e-18 or 1 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0 56.7%

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

      1. mul-1-neg 56.7%

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

      2. distribute-neg-frac2 56.7%

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

      3. neg-sub0 56.7%

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

      4. associate--r- 56.7%

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

      5. metadata-eval 56.7%

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

   5. Simplified 56.7%

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

   6. Taylor expanded in z around inf 52.6%

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

      1. *-commutative 52.6%

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

      2. *-lft-identity 52.6%

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

      3. times-frac 55.9%

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

      4. /-rgt-identity 55.9%

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

   8. Simplified 55.9%

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

   if -2.3000000000000001e-18 < z < 1

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0 35.5%

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

      1. mul-1-neg 35.5%

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

      2. distribute-neg-frac2 35.5%

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

      3. neg-sub0 35.5%

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

      4. associate--r- 35.5%

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

      5. metadata-eval 35.5%

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

   5. Simplified 35.5%

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

   6. Taylor expanded in z around 0 34.8%

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

      1. *-commutative 34.8%

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

      2. neg-mul-1 34.8%

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

      3. distribute-lft-neg-in 34.8%

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

   8. Simplified 34.8%

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

4. Final simplification 46.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{-18} \lor \neg \left(z \leq 1\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(-t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 22.6% accurate, 2.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

3. Taylor expanded in y around 0 46.8%

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

   1. mul-1-neg 46.8%

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

   2. distribute-neg-frac2 46.8%

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

   3. neg-sub0 46.8%

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

   4. associate--r- 46.8%

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

   5. metadata-eval 46.8%

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

5. Simplified 46.8%

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

6. Taylor expanded in z around 0 23.8%

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

   1. *-commutative 23.8%

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

   2. neg-mul-1 23.8%

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

   3. distribute-lft-neg-in 23.8%

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

8. Simplified 23.8%

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

9. Final simplification 23.8%

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

Developer target: 94.7% 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 2024076 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :alt
  (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) -7.623226303312042e-196) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z))))) (if (< (* x (- (/ y z) (/ t (- 1.0 z)))) 1.4133944927702302e-211) (+ (/ (* y x) z) (- (/ (* t x) (- 1.0 z)))) (* x (- (/ y z) (* t (/ 1.0 (- 1.0 z)))))))

  (* x (- (/ y z) (/ t (- 1.0 z)))))