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

Percentage Accurate: 94.2% → 99.0%

Time: 8.4s

Alternatives: 11

Speedup: 0.2×

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

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 94.2% 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: 99.0% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} - \frac{t}{1 - z}\\ t_2 := t_1 \cdot x\\ \mathbf{if}\;t_1 \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t_1 \leq -4 \cdot 10^{-197}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t_1 \leq 0:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{elif}\;t_1 \leq 2 \cdot 10^{+290}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ y z) (/ t (- 1.0 z)))) (t_2 (* t_1 x)))
   (if (<= t_1 (- INFINITY))
     (* y (/ x z))
     (if (<= t_1 -4e-197)
       t_2
       (if (<= t_1 0.0)
         (* (/ x z) (+ y t))
         (if (<= t_1 2e+290) t_2 (/ y (/ z x))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double t_2 = t_1 * x;
	double tmp;
	if (t_1 <= -((double) INFINITY)) {
		tmp = y * (x / z);
	} else if (t_1 <= -4e-197) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (x / z) * (y + t);
	} else if (t_1 <= 2e+290) {
		tmp = t_2;
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (y / z) - (t / (1.0 - z));
	double t_2 = t_1 * x;
	double tmp;
	if (t_1 <= -Double.POSITIVE_INFINITY) {
		tmp = y * (x / z);
	} else if (t_1 <= -4e-197) {
		tmp = t_2;
	} else if (t_1 <= 0.0) {
		tmp = (x / z) * (y + t);
	} else if (t_1 <= 2e+290) {
		tmp = t_2;
	} else {
		tmp = y / (z / x);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (y / z) - (t / (1.0 - z))
	t_2 = t_1 * x
	tmp = 0
	if t_1 <= -math.inf:
		tmp = y * (x / z)
	elif t_1 <= -4e-197:
		tmp = t_2
	elif t_1 <= 0.0:
		tmp = (x / z) * (y + t)
	elif t_1 <= 2e+290:
		tmp = t_2
	else:
		tmp = y / (z / x)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y / z) - Float64(t / Float64(1.0 - z)))
	t_2 = Float64(t_1 * x)
	tmp = 0.0
	if (t_1 <= Float64(-Inf))
		tmp = Float64(y * Float64(x / z));
	elseif (t_1 <= -4e-197)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = Float64(Float64(x / z) * Float64(y + t));
	elseif (t_1 <= 2e+290)
		tmp = t_2;
	else
		tmp = Float64(y / Float64(z / x));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (y / z) - (t / (1.0 - z));
	t_2 = t_1 * x;
	tmp = 0.0;
	if (t_1 <= -Inf)
		tmp = y * (x / z);
	elseif (t_1 <= -4e-197)
		tmp = t_2;
	elseif (t_1 <= 0.0)
		tmp = (x / z) * (y + t);
	elseif (t_1 <= 2e+290)
		tmp = t_2;
	else
		tmp = y / (z / x);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] - N[(t / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * x), $MachinePrecision]}, If[LessEqual[t$95$1, (-Infinity)], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, -4e-197], t$95$2, If[LessEqual[t$95$1, 0.0], N[(N[(x / z), $MachinePrecision] * N[(y + t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e+290], t$95$2, N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -inf.0

   1. Initial program 59.9%

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

      1. sub-neg 59.9%

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

      2. distribute-rgt-in 49.4%

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

      3. fma-def 49.4%

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

      4. distribute-neg-frac 49.4%

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

   3. Applied egg-rr 49.4%

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

   4. Taylor expanded in y around inf 99.8%

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

      1. associate-*r/ 99.8%

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

   6. Simplified 99.8%

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

   if -inf.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -3.9999999999999999e-197 or -0.0 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < 2.00000000000000012e290

   1. Initial program 99.6%

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

   if -3.9999999999999999e-197 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z))) < -0.0

   1. Initial program 57.0%

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

   2. Taylor expanded in z around inf 99.5%

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

      1. *-commutative 99.5%

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

      2. associate-/l* 53.2%

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

      3. associate-/r/ 99.7%

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

      4. cancel-sign-sub-inv 99.7%

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

      5. metadata-eval 99.7%

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

      6. *-lft-identity 99.7%

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

   4. Simplified 99.7%

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

   if 2.00000000000000012e290 < (-.f64 (/.f64 y z) (/.f64 t (-.f64 1 z)))

   1. Initial program 60.3%

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

   2. Taylor expanded in y around inf 94.7%

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

      1. associate-*l/ 55.3%

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

   4. Simplified 55.3%

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

      1. associate-*l/ 94.7%

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

      2. associate-*r/ 99.7%

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

      3. clear-num 99.7%

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

      4. un-div-inv 99.9%

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

   6. Applied egg-rr 99.9%

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

4. Final simplification 99.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{y}{z} - \frac{t}{1 - z} \leq -\infty:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq -4 \cdot 10^{-197}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 0:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{elif}\;\frac{y}{z} - \frac{t}{1 - z} \leq 2 \cdot 10^{+290}:\\ \;\;\;\;\left(\frac{y}{z} - \frac{t}{1 - z}\right) \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \end{array} \]

Alternative 2: 71.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+163}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+158}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-203}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+90}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))) (t_2 (* x (- (/ y z) t))))
   (if (<= z -2.1e+177)
     t_1
     (if (<= z -2e+163)
       (/ y (/ z x))
       (if (<= z -1.2e+158)
         t_1
         (if (<= z -7.5e-203)
           t_2
           (if (<= z -7.5e-307)
             (/ (* y x) z)
             (if (<= z 6.5e+90) t_2 (/ x (/ z t))))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double t_2 = x * ((y / z) - t);
	double tmp;
	if (z <= -2.1e+177) {
		tmp = t_1;
	} else if (z <= -2e+163) {
		tmp = y / (z / x);
	} else if (z <= -1.2e+158) {
		tmp = t_1;
	} else if (z <= -7.5e-203) {
		tmp = t_2;
	} else if (z <= -7.5e-307) {
		tmp = (y * x) / z;
	} else if (z <= 6.5e+90) {
		tmp = t_2;
	} else {
		tmp = x / (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) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = x * (t / z)
    t_2 = x * ((y / z) - t)
    if (z <= (-2.1d+177)) then
        tmp = t_1
    else if (z <= (-2d+163)) then
        tmp = y / (z / x)
    else if (z <= (-1.2d+158)) then
        tmp = t_1
    else if (z <= (-7.5d-203)) then
        tmp = t_2
    else if (z <= (-7.5d-307)) then
        tmp = (y * x) / z
    else if (z <= 6.5d+90) then
        tmp = t_2
    else
        tmp = x / (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double t_2 = x * ((y / z) - t);
	double tmp;
	if (z <= -2.1e+177) {
		tmp = t_1;
	} else if (z <= -2e+163) {
		tmp = y / (z / x);
	} else if (z <= -1.2e+158) {
		tmp = t_1;
	} else if (z <= -7.5e-203) {
		tmp = t_2;
	} else if (z <= -7.5e-307) {
		tmp = (y * x) / z;
	} else if (z <= 6.5e+90) {
		tmp = t_2;
	} else {
		tmp = x / (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / z)
	t_2 = x * ((y / z) - t)
	tmp = 0
	if z <= -2.1e+177:
		tmp = t_1
	elif z <= -2e+163:
		tmp = y / (z / x)
	elif z <= -1.2e+158:
		tmp = t_1
	elif z <= -7.5e-203:
		tmp = t_2
	elif z <= -7.5e-307:
		tmp = (y * x) / z
	elif z <= 6.5e+90:
		tmp = t_2
	else:
		tmp = x / (z / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / z))
	t_2 = Float64(x * Float64(Float64(y / z) - t))
	tmp = 0.0
	if (z <= -2.1e+177)
		tmp = t_1;
	elseif (z <= -2e+163)
		tmp = Float64(y / Float64(z / x));
	elseif (z <= -1.2e+158)
		tmp = t_1;
	elseif (z <= -7.5e-203)
		tmp = t_2;
	elseif (z <= -7.5e-307)
		tmp = Float64(Float64(y * x) / z);
	elseif (z <= 6.5e+90)
		tmp = t_2;
	else
		tmp = Float64(x / Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / z);
	t_2 = x * ((y / z) - t);
	tmp = 0.0;
	if (z <= -2.1e+177)
		tmp = t_1;
	elseif (z <= -2e+163)
		tmp = y / (z / x);
	elseif (z <= -1.2e+158)
		tmp = t_1;
	elseif (z <= -7.5e-203)
		tmp = t_2;
	elseif (z <= -7.5e-307)
		tmp = (y * x) / z;
	elseif (z <= 6.5e+90)
		tmp = t_2;
	else
		tmp = x / (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e+177], t$95$1, If[LessEqual[z, -2e+163], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.2e+158], t$95$1, If[LessEqual[z, -7.5e-203], t$95$2, If[LessEqual[z, -7.5e-307], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 6.5e+90], t$95$2, N[(x / N[(z / t), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.10000000000000013e177 or -1.9999999999999999e163 < z < -1.20000000000000004e158

   1. Initial program 84.7%

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

   2. Taylor expanded in z around inf 79.4%

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

      1. *-commutative 79.4%

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

      2. associate-/l* 82.7%

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

      3. associate-/r/ 82.9%

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

      4. cancel-sign-sub-inv 82.9%

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

      5. metadata-eval 82.9%

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

      6. *-lft-identity 82.9%

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

   4. Simplified 82.9%

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

   5. Taylor expanded in x around 0 79.4%

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

   6. Taylor expanded in y around 0 57.0%

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

      1. associate-/l* 56.6%

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

      2. associate-/r/ 64.0%

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

   8. Simplified 64.0%

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

   if -2.10000000000000013e177 < z < -1.9999999999999999e163

   1. Initial program 86.3%

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

   2. Taylor expanded in y around inf 72.3%

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

      1. associate-*l/ 86.3%

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

   4. Simplified 86.3%

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

      1. associate-*l/ 72.3%

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

      2. associate-*r/ 86.1%

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

      3. clear-num 86.1%

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

      4. un-div-inv 86.3%

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

   6. Applied egg-rr 86.3%

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

   if -1.20000000000000004e158 < z < -7.50000000000000027e-203 or -7.5000000000000006e-307 < z < 6.5000000000000001e90

   1. Initial program 94.2%

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

   2. Taylor expanded in z around 0 79.2%

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

      1. associate-*l/ 78.5%

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

      2. associate-*r* 78.5%

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

      3. neg-mul-1 78.5%

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

      4. distribute-rgt-out 82.2%

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

      5. unsub-neg 82.2%

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

   4. Simplified 82.2%

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

   if -7.50000000000000027e-203 < z < -7.5000000000000006e-307

   1. Initial program 74.5%

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

   2. Taylor expanded in y around inf 90.2%

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

   if 6.5000000000000001e90 < z

   1. Initial program 95.8%

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

   2. Taylor expanded in z around inf 84.1%

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

      1. *-commutative 84.1%

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

      2. associate-/l* 95.8%

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

      3. neg-mul-1 95.8%

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

   4. Simplified 95.8%

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

   5. Taylor expanded in y around 0 68.8%

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{t}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+177}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq -2 \cdot 10^{+163}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;z \leq -1.2 \cdot 10^{+158}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-203}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-307}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{+90}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]

Alternative 3: 70.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z + -1}\\ \mathbf{if}\;y \leq -7.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-307}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-232}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-128}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-72}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+46}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t (+ z -1.0)))))
   (if (<= y -7.5e-78)
     (/ y (/ z x))
     (if (<= y 2.45e-307)
       t_1
       (if (<= y 9.8e-232)
         (* x (- (/ y z) t))
         (if (<= y 8.2e-128)
           t_1
           (if (<= y 2.6e-72)
             (* y (/ x z))
             (if (<= y 1.2e+46) t_1 (/ (* y x) z)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / (z + -1.0));
	double tmp;
	if (y <= -7.5e-78) {
		tmp = y / (z / x);
	} else if (y <= 2.45e-307) {
		tmp = t_1;
	} else if (y <= 9.8e-232) {
		tmp = x * ((y / z) - t);
	} else if (y <= 8.2e-128) {
		tmp = t_1;
	} else if (y <= 2.6e-72) {
		tmp = y * (x / z);
	} else if (y <= 1.2e+46) {
		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 = x * (t / (z + (-1.0d0)))
    if (y <= (-7.5d-78)) then
        tmp = y / (z / x)
    else if (y <= 2.45d-307) then
        tmp = t_1
    else if (y <= 9.8d-232) then
        tmp = x * ((y / z) - t)
    else if (y <= 8.2d-128) then
        tmp = t_1
    else if (y <= 2.6d-72) then
        tmp = y * (x / z)
    else if (y <= 1.2d+46) 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 = x * (t / (z + -1.0));
	double tmp;
	if (y <= -7.5e-78) {
		tmp = y / (z / x);
	} else if (y <= 2.45e-307) {
		tmp = t_1;
	} else if (y <= 9.8e-232) {
		tmp = x * ((y / z) - t);
	} else if (y <= 8.2e-128) {
		tmp = t_1;
	} else if (y <= 2.6e-72) {
		tmp = y * (x / z);
	} else if (y <= 1.2e+46) {
		tmp = t_1;
	} else {
		tmp = (y * x) / z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / (z + -1.0))
	tmp = 0
	if y <= -7.5e-78:
		tmp = y / (z / x)
	elif y <= 2.45e-307:
		tmp = t_1
	elif y <= 9.8e-232:
		tmp = x * ((y / z) - t)
	elif y <= 8.2e-128:
		tmp = t_1
	elif y <= 2.6e-72:
		tmp = y * (x / z)
	elif y <= 1.2e+46:
		tmp = t_1
	else:
		tmp = (y * x) / z
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / Float64(z + -1.0)))
	tmp = 0.0
	if (y <= -7.5e-78)
		tmp = Float64(y / Float64(z / x));
	elseif (y <= 2.45e-307)
		tmp = t_1;
	elseif (y <= 9.8e-232)
		tmp = Float64(x * Float64(Float64(y / z) - t));
	elseif (y <= 8.2e-128)
		tmp = t_1;
	elseif (y <= 2.6e-72)
		tmp = Float64(y * Float64(x / z));
	elseif (y <= 1.2e+46)
		tmp = t_1;
	else
		tmp = Float64(Float64(y * x) / z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / (z + -1.0));
	tmp = 0.0;
	if (y <= -7.5e-78)
		tmp = y / (z / x);
	elseif (y <= 2.45e-307)
		tmp = t_1;
	elseif (y <= 9.8e-232)
		tmp = x * ((y / z) - t);
	elseif (y <= 8.2e-128)
		tmp = t_1;
	elseif (y <= 2.6e-72)
		tmp = y * (x / z);
	elseif (y <= 1.2e+46)
		tmp = t_1;
	else
		tmp = (y * x) / z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t / N[(z + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -7.5e-78], N[(y / N[(z / x), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.45e-307], t$95$1, If[LessEqual[y, 9.8e-232], N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 8.2e-128], t$95$1, If[LessEqual[y, 2.6e-72], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+46], t$95$1, N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision]]]]]]]]

Derivation

1. Split input into 5 regimes

2. if y < -7.50000000000000041e-78

   1. Initial program 90.4%

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

   2. Taylor expanded in y around inf 77.5%

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

      1. associate-*l/ 71.8%

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

   4. Simplified 71.8%

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

      1. associate-*l/ 77.5%

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

      2. associate-*r/ 78.7%

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

      3. clear-num 78.7%

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

      4. un-div-inv 78.8%

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

   6. Applied egg-rr 78.8%

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

   if -7.50000000000000041e-78 < y < 2.4500000000000001e-307 or 9.8000000000000006e-232 < y < 8.1999999999999999e-128 or 2.59999999999999996e-72 < y < 1.20000000000000004e46

   1. Initial program 94.4%

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

   2. Taylor expanded in y around 0 70.9%

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

      1. associate-*r/ 70.9%

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

      2. associate-*r* 70.9%

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

      3. neg-mul-1 70.9%

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

      4. associate-*l/ 74.8%

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

      5. *-commutative 74.8%

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

      6. neg-mul-1 74.8%

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

      7. *-commutative 74.8%

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

      8. associate-*r/ 74.9%

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

      9. metadata-eval 74.9%

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

      10. associate-/r* 74.9%

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

      11. neg-mul-1 74.9%

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

      12. associate-*r/ 74.8%

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

      13. *-rgt-identity 74.8%

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

      14. neg-sub0 74.8%

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

      15. associate--r- 74.8%

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

      16. metadata-eval 74.8%

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

   4. Simplified 74.8%

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

   if 2.4500000000000001e-307 < y < 9.8000000000000006e-232

   1. Initial program 99.9%

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

   2. Taylor expanded in z around 0 72.2%

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

      1. associate-*l/ 86.0%

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

      2. associate-*r* 86.0%

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

      3. neg-mul-1 86.0%

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

      4. distribute-rgt-out 86.0%

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

      5. unsub-neg 86.0%

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

   4. Simplified 86.0%

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

   if 8.1999999999999999e-128 < y < 2.59999999999999996e-72

   1. Initial program 88.1%

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

      1. sub-neg 88.1%

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

      2. distribute-rgt-in 88.1%

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

      3. fma-def 88.1%

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

      4. distribute-neg-frac 88.1%

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

   3. Applied egg-rr 88.1%

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

   4. Taylor expanded in y around inf 72.7%

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

      1. associate-*r/ 72.7%

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

   6. Simplified 72.7%

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

   if 1.20000000000000004e46 < y

   1. Initial program 82.3%

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

   2. Taylor expanded in y around inf 88.5%

      \[\leadsto \color{blue}{\frac{y \cdot x}{z}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.5 \cdot 10^{-78}:\\ \;\;\;\;\frac{y}{\frac{z}{x}}\\ \mathbf{elif}\;y \leq 2.45 \cdot 10^{-307}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{elif}\;y \leq 9.8 \cdot 10^{-232}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{-128}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{elif}\;y \leq 2.6 \cdot 10^{-72}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+46}:\\ \;\;\;\;x \cdot \frac{t}{z + -1}\\ \mathbf{else}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \end{array} \]

Alternative 4: 88.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := \frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-302}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) t))) (t_2 (* (/ x z) (+ y t))))
   (if (<= z -1.0)
     t_2
     (if (<= z -7.5e-203)
       t_1
       (if (<= z 8.8e-302) (/ (* y x) z) (if (<= z 4.6e-6) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - t);
	double t_2 = (x / z) * (y + t);
	double tmp;
	if (z <= -1.0) {
		tmp = t_2;
	} else if (z <= -7.5e-203) {
		tmp = t_1;
	} else if (z <= 8.8e-302) {
		tmp = (y * x) / z;
	} else if (z <= 4.6e-6) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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)
    t_2 = (x / z) * (y + t)
    if (z <= (-1.0d0)) then
        tmp = t_2
    else if (z <= (-7.5d-203)) then
        tmp = t_1
    else if (z <= 8.8d-302) then
        tmp = (y * x) / z
    else if (z <= 4.6d-6) then
        tmp = t_1
    else
        tmp = t_2
    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);
	double t_2 = (x / z) * (y + t);
	double tmp;
	if (z <= -1.0) {
		tmp = t_2;
	} else if (z <= -7.5e-203) {
		tmp = t_1;
	} else if (z <= 8.8e-302) {
		tmp = (y * x) / z;
	} else if (z <= 4.6e-6) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) - t)
	t_2 = (x / z) * (y + t)
	tmp = 0
	if z <= -1.0:
		tmp = t_2
	elif z <= -7.5e-203:
		tmp = t_1
	elif z <= 8.8e-302:
		tmp = (y * x) / z
	elif z <= 4.6e-6:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y / z) - t))
	t_2 = Float64(Float64(x / z) * Float64(y + t))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_2;
	elseif (z <= -7.5e-203)
		tmp = t_1;
	elseif (z <= 8.8e-302)
		tmp = Float64(Float64(y * x) / z);
	elseif (z <= 4.6e-6)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) - t);
	t_2 = (x / z) * (y + t);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_2;
	elseif (z <= -7.5e-203)
		tmp = t_1;
	elseif (z <= 8.8e-302)
		tmp = (y * x) / z;
	elseif (z <= 4.6e-6)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / z), $MachinePrecision] * N[(y + t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$2, If[LessEqual[z, -7.5e-203], t$95$1, If[LessEqual[z, 8.8e-302], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 4.6e-6], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 4.6e-6 < z

   1. Initial program 94.1%

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

   2. Taylor expanded in z around inf 87.5%

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

      1. *-commutative 87.5%

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

      2. associate-/l* 93.1%

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

      3. associate-/r/ 86.3%

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

      4. cancel-sign-sub-inv 86.3%

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

      5. metadata-eval 86.3%

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

      6. *-lft-identity 86.3%

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

   4. Simplified 86.3%

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

   if -1 < z < -7.50000000000000027e-203 or 8.8000000000000003e-302 < z < 4.6e-6

   1. Initial program 91.1%

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

   2. Taylor expanded in z around 0 86.6%

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

      1. associate-*l/ 84.4%

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

      2. associate-*r* 84.4%

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

      3. neg-mul-1 84.4%

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

      4. distribute-rgt-out 90.2%

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

      5. unsub-neg 90.2%

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

   4. Simplified 90.2%

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

   if -7.50000000000000027e-203 < z < 8.8000000000000003e-302

   1. Initial program 74.5%

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

   2. Taylor expanded in y around inf 90.2%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \mathbf{elif}\;z \leq -7.5 \cdot 10^{-203}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-302}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{z} \cdot \left(y + t\right)\\ \end{array} \]

Alternative 5: 92.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \left(\frac{y}{z} - t\right)\\ t_2 := x \cdot \frac{y + t}{z}\\ \mathbf{if}\;z \leq -1:\\ \;\;\;\;t_2\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-203}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-305}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (- (/ y z) t))) (t_2 (* x (/ (+ y t) z))))
   (if (<= z -1.0)
     t_2
     (if (<= z -1.25e-203)
       t_1
       (if (<= z 1.35e-305) (/ (* y x) z) (if (<= z 4.6e-6) t_1 t_2))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * ((y / z) - t);
	double t_2 = x * ((y + t) / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_2;
	} else if (z <= -1.25e-203) {
		tmp = t_1;
	} else if (z <= 1.35e-305) {
		tmp = (y * x) / z;
	} else if (z <= 4.6e-6) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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)
    t_2 = x * ((y + t) / z)
    if (z <= (-1.0d0)) then
        tmp = t_2
    else if (z <= (-1.25d-203)) then
        tmp = t_1
    else if (z <= 1.35d-305) then
        tmp = (y * x) / z
    else if (z <= 4.6d-6) then
        tmp = t_1
    else
        tmp = t_2
    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);
	double t_2 = x * ((y + t) / z);
	double tmp;
	if (z <= -1.0) {
		tmp = t_2;
	} else if (z <= -1.25e-203) {
		tmp = t_1;
	} else if (z <= 1.35e-305) {
		tmp = (y * x) / z;
	} else if (z <= 4.6e-6) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * ((y / z) - t)
	t_2 = x * ((y + t) / z)
	tmp = 0
	if z <= -1.0:
		tmp = t_2
	elif z <= -1.25e-203:
		tmp = t_1
	elif z <= 1.35e-305:
		tmp = (y * x) / z
	elif z <= 4.6e-6:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(Float64(y / z) - t))
	t_2 = Float64(x * Float64(Float64(y + t) / z))
	tmp = 0.0
	if (z <= -1.0)
		tmp = t_2;
	elseif (z <= -1.25e-203)
		tmp = t_1;
	elseif (z <= 1.35e-305)
		tmp = Float64(Float64(y * x) / z);
	elseif (z <= 4.6e-6)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * ((y / z) - t);
	t_2 = x * ((y + t) / z);
	tmp = 0.0;
	if (z <= -1.0)
		tmp = t_2;
	elseif (z <= -1.25e-203)
		tmp = t_1;
	elseif (z <= 1.35e-305)
		tmp = (y * x) / z;
	elseif (z <= 4.6e-6)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(N[(y / z), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(N[(y + t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.0], t$95$2, If[LessEqual[z, -1.25e-203], t$95$1, If[LessEqual[z, 1.35e-305], N[(N[(y * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 4.6e-6], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1 or 4.6e-6 < z

   1. Initial program 94.1%

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

   2. Taylor expanded in z around inf 87.5%

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

      1. associate-/l* 85.8%

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

      2. associate-/r/ 93.5%

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

      3. cancel-sign-sub-inv 93.5%

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

      4. metadata-eval 93.5%

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

      5. *-lft-identity 93.5%

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

   4. Simplified 93.5%

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

   if -1 < z < -1.25e-203 or 1.35e-305 < z < 4.6e-6

   1. Initial program 91.1%

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

   2. Taylor expanded in z around 0 86.6%

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

      1. associate-*l/ 84.4%

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

      2. associate-*r* 84.4%

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

      3. neg-mul-1 84.4%

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

      4. distribute-rgt-out 90.2%

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

      5. unsub-neg 90.2%

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

   4. Simplified 90.2%

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

   if -1.25e-203 < z < 1.35e-305

   1. Initial program 74.5%

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

   2. Taylor expanded in y around inf 90.2%

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

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \mathbf{elif}\;z \leq -1.25 \cdot 10^{-203}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-305}:\\ \;\;\;\;\frac{y \cdot x}{z}\\ \mathbf{elif}\;z \leq 4.6 \cdot 10^{-6}:\\ \;\;\;\;x \cdot \left(\frac{y}{z} - t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y + t}{z}\\ \end{array} \]

Alternative 6: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{+107} \lor \neg \left(t \leq -2.2\right) \land \left(t \leq -1.85 \cdot 10^{-13} \lor \neg \left(t \leq 1.4 \cdot 10^{+172}\right)\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= t -2.35e+107)
         (and (not (<= t -2.2)) (or (<= t -1.85e-13) (not (<= t 1.4e+172)))))
   (* x (/ t z))
   (* y (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((t <= -2.35e+107) || (!(t <= -2.2) && ((t <= -1.85e-13) || !(t <= 1.4e+172)))) {
		tmp = x * (t / z);
	} 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 ((t <= (-2.35d+107)) .or. (.not. (t <= (-2.2d0))) .and. (t <= (-1.85d-13)) .or. (.not. (t <= 1.4d+172))) then
        tmp = x * (t / z)
    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 ((t <= -2.35e+107) || (!(t <= -2.2) && ((t <= -1.85e-13) || !(t <= 1.4e+172)))) {
		tmp = x * (t / z);
	} else {
		tmp = y * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (t <= -2.35e+107) or (not (t <= -2.2) and ((t <= -1.85e-13) or not (t <= 1.4e+172))):
		tmp = x * (t / z)
	else:
		tmp = y * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((t <= -2.35e+107) || (!(t <= -2.2) && ((t <= -1.85e-13) || !(t <= 1.4e+172))))
		tmp = Float64(x * Float64(t / z));
	else
		tmp = Float64(y * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((t <= -2.35e+107) || (~((t <= -2.2)) && ((t <= -1.85e-13) || ~((t <= 1.4e+172)))))
		tmp = x * (t / z);
	else
		tmp = y * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[t, -2.35e+107], And[N[Not[LessEqual[t, -2.2]], $MachinePrecision], Or[LessEqual[t, -1.85e-13], N[Not[LessEqual[t, 1.4e+172]], $MachinePrecision]]]], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -2.35e107 or -2.2000000000000002 < t < -1.84999999999999994e-13 or 1.4e172 < t

   1. Initial program 94.0%

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

   2. Taylor expanded in z around inf 59.8%

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

      1. *-commutative 59.8%

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

      2. associate-/l* 70.1%

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

      3. associate-/r/ 61.1%

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

      4. cancel-sign-sub-inv 61.1%

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

      5. metadata-eval 61.1%

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

      6. *-lft-identity 61.1%

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

   4. Simplified 61.1%

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

   5. Taylor expanded in x around 0 59.8%

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

   6. Taylor expanded in y around 0 52.8%

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

      1. associate-/l* 55.0%

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

      2. associate-/r/ 62.9%

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

   8. Simplified 62.9%

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

   if -2.35e107 < t < -2.2000000000000002 or -1.84999999999999994e-13 < t < 1.4e172

   1. Initial program 89.3%

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

      1. sub-neg 89.3%

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

      2. distribute-rgt-in 89.2%

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

      3. fma-def 89.3%

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

      4. distribute-neg-frac 89.3%

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

   3. Applied egg-rr 89.3%

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

   4. Taylor expanded in y around inf 73.0%

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

      1. associate-*r/ 73.6%

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

   6. Simplified 73.6%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.35 \cdot 10^{+107} \lor \neg \left(t \leq -2.2\right) \land \left(t \leq -1.85 \cdot 10^{-13} \lor \neg \left(t \leq 1.4 \cdot 10^{+172}\right)\right):\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \end{array} \]

Alternative 7: 67.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ t_2 := y \cdot \frac{x}{z}\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{+105}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -1.65:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+172}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))) (t_2 (* y (/ x z))))
   (if (<= t -1.45e+105)
     t_1
     (if (<= t -1.65)
       t_2
       (if (<= t -1.85e-13) (* t (/ x z)) (if (<= t 1.45e+172) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double t_2 = y * (x / z);
	double tmp;
	if (t <= -1.45e+105) {
		tmp = t_1;
	} else if (t <= -1.65) {
		tmp = t_2;
	} else if (t <= -1.85e-13) {
		tmp = t * (x / z);
	} else if (t <= 1.45e+172) {
		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 * (t / z)
    t_2 = y * (x / z)
    if (t <= (-1.45d+105)) then
        tmp = t_1
    else if (t <= (-1.65d0)) then
        tmp = t_2
    else if (t <= (-1.85d-13)) then
        tmp = t * (x / z)
    else if (t <= 1.45d+172) 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 * (t / z);
	double t_2 = y * (x / z);
	double tmp;
	if (t <= -1.45e+105) {
		tmp = t_1;
	} else if (t <= -1.65) {
		tmp = t_2;
	} else if (t <= -1.85e-13) {
		tmp = t * (x / z);
	} else if (t <= 1.45e+172) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / z)
	t_2 = y * (x / z)
	tmp = 0
	if t <= -1.45e+105:
		tmp = t_1
	elif t <= -1.65:
		tmp = t_2
	elif t <= -1.85e-13:
		tmp = t * (x / z)
	elif t <= 1.45e+172:
		tmp = t_2
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / z))
	t_2 = Float64(y * Float64(x / z))
	tmp = 0.0
	if (t <= -1.45e+105)
		tmp = t_1;
	elseif (t <= -1.65)
		tmp = t_2;
	elseif (t <= -1.85e-13)
		tmp = Float64(t * Float64(x / z));
	elseif (t <= 1.45e+172)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / z);
	t_2 = y * (x / z);
	tmp = 0.0;
	if (t <= -1.45e+105)
		tmp = t_1;
	elseif (t <= -1.65)
		tmp = t_2;
	elseif (t <= -1.85e-13)
		tmp = t * (x / z);
	elseif (t <= 1.45e+172)
		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[(t / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1.45e+105], t$95$1, If[LessEqual[t, -1.65], t$95$2, If[LessEqual[t, -1.85e-13], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 1.45e+172], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.45000000000000005e105 or 1.45e172 < t

   1. Initial program 93.5%

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

   2. Taylor expanded in z around inf 56.7%

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

      1. *-commutative 56.7%

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

      2. associate-/l* 67.8%

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

      3. associate-/r/ 59.2%

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

      4. cancel-sign-sub-inv 59.2%

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

      5. metadata-eval 59.2%

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

      6. *-lft-identity 59.2%

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

   4. Simplified 59.2%

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

   5. Taylor expanded in x around 0 56.7%

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

   6. Taylor expanded in y around 0 50.4%

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

      1. associate-/l* 52.7%

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

      2. associate-/r/ 61.2%

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

   8. Simplified 61.2%

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

   if -1.45000000000000005e105 < t < -1.6499999999999999 or -1.84999999999999994e-13 < t < 1.45e172

   1. Initial program 89.3%

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

      1. sub-neg 89.3%

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

      2. distribute-rgt-in 89.2%

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

      3. fma-def 89.3%

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

      4. distribute-neg-frac 89.3%

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

   3. Applied egg-rr 89.3%

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

   4. Taylor expanded in y around inf 73.0%

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

      1. associate-*r/ 73.6%

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

   6. Simplified 73.6%

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

   if -1.6499999999999999 < t < -1.84999999999999994e-13

   1. Initial program 99.5%

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

   2. Taylor expanded in z around inf 99.0%

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

      1. *-commutative 99.0%

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

      2. associate-/l* 99.7%

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

      3. neg-mul-1 99.7%

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

   4. Simplified 99.7%

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

   5. Taylor expanded in y around 0 84.2%

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

      1. associate-/r/ 84.2%

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

   7. Applied egg-rr 84.2%

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

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.45 \cdot 10^{+105}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;t \leq -1.65:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq -1.85 \cdot 10^{-13}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{+172}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 8: 62.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{t}{z}\\ \mathbf{if}\;z \leq -2.1 \cdot 10^{+177}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+15}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+90}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ t z))))
   (if (<= z -2.1e+177)
     t_1
     (if (<= z -1.02e+55)
       (* (/ y z) x)
       (if (<= z -1.15e+15)
         (* t (/ x z))
         (if (<= z 5.8e+90) (* y (/ x z)) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (z <= -2.1e+177) {
		tmp = t_1;
	} else if (z <= -1.02e+55) {
		tmp = (y / z) * x;
	} else if (z <= -1.15e+15) {
		tmp = t * (x / z);
	} else if (z <= 5.8e+90) {
		tmp = y * (x / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (t / z)
    if (z <= (-2.1d+177)) then
        tmp = t_1
    else if (z <= (-1.02d+55)) then
        tmp = (y / z) * x
    else if (z <= (-1.15d+15)) then
        tmp = t * (x / z)
    else if (z <= 5.8d+90) then
        tmp = y * (x / z)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x * (t / z);
	double tmp;
	if (z <= -2.1e+177) {
		tmp = t_1;
	} else if (z <= -1.02e+55) {
		tmp = (y / z) * x;
	} else if (z <= -1.15e+15) {
		tmp = t * (x / z);
	} else if (z <= 5.8e+90) {
		tmp = y * (x / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (t / z)
	tmp = 0
	if z <= -2.1e+177:
		tmp = t_1
	elif z <= -1.02e+55:
		tmp = (y / z) * x
	elif z <= -1.15e+15:
		tmp = t * (x / z)
	elif z <= 5.8e+90:
		tmp = y * (x / z)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(t / z))
	tmp = 0.0
	if (z <= -2.1e+177)
		tmp = t_1;
	elseif (z <= -1.02e+55)
		tmp = Float64(Float64(y / z) * x);
	elseif (z <= -1.15e+15)
		tmp = Float64(t * Float64(x / z));
	elseif (z <= 5.8e+90)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (t / z);
	tmp = 0.0;
	if (z <= -2.1e+177)
		tmp = t_1;
	elseif (z <= -1.02e+55)
		tmp = (y / z) * x;
	elseif (z <= -1.15e+15)
		tmp = t * (x / z);
	elseif (z <= 5.8e+90)
		tmp = y * (x / z);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.1e+177], t$95$1, If[LessEqual[z, -1.02e+55], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, -1.15e+15], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 5.8e+90], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -2.10000000000000013e177 or 5.8000000000000003e90 < z

   1. Initial program 91.2%

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

   2. Taylor expanded in z around inf 82.0%

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

      1. *-commutative 82.0%

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

      2. associate-/l* 90.4%

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

      3. associate-/r/ 82.4%

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

      4. cancel-sign-sub-inv 82.4%

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

      5. metadata-eval 82.4%

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

      6. *-lft-identity 82.4%

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

   4. Simplified 82.4%

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

   5. Taylor expanded in x around 0 82.0%

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

   6. Taylor expanded in y around 0 58.9%

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

      1. associate-/l* 57.9%

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

      2. associate-/r/ 66.5%

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

   8. Simplified 66.5%

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

   if -2.10000000000000013e177 < z < -1.02000000000000002e55

   1. Initial program 95.2%

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

   2. Taylor expanded in y around inf 60.2%

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

      1. associate-*l/ 68.8%

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

   4. Simplified 68.8%

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

   if -1.02000000000000002e55 < z < -1.15e15

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. neg-mul-1 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around 0 75.6%

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

      1. associate-/r/ 75.7%

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

   7. Applied egg-rr 75.7%

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

   if -1.15e15 < z < 5.8000000000000003e90

   1. Initial program 89.1%

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

      1. sub-neg 89.1%

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

      2. distribute-rgt-in 82.7%

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

      3. fma-def 84.8%

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

      4. distribute-neg-frac 84.8%

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

   3. Applied egg-rr 84.8%

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

   4. Taylor expanded in y around inf 70.6%

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

      1. associate-*r/ 72.0%

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

   6. Simplified 72.0%

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

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.1 \cdot 10^{+177}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -1.15 \cdot 10^{+15}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+90}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]

Alternative 9: 62.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+177}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -26000000000000:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.3e+177)
   (* x (/ t z))
   (if (<= z -3.9e+55)
     (* (/ y z) x)
     (if (<= z -26000000000000.0)
       (* t (/ x z))
       (if (<= z 1.15e+104) (* y (/ x z)) (/ x (/ z t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.3e+177) {
		tmp = x * (t / z);
	} else if (z <= -3.9e+55) {
		tmp = (y / z) * x;
	} else if (z <= -26000000000000.0) {
		tmp = t * (x / z);
	} else if (z <= 1.15e+104) {
		tmp = y * (x / z);
	} else {
		tmp = x / (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 <= (-2.3d+177)) then
        tmp = x * (t / z)
    else if (z <= (-3.9d+55)) then
        tmp = (y / z) * x
    else if (z <= (-26000000000000.0d0)) then
        tmp = t * (x / z)
    else if (z <= 1.15d+104) then
        tmp = y * (x / z)
    else
        tmp = x / (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 <= -2.3e+177) {
		tmp = x * (t / z);
	} else if (z <= -3.9e+55) {
		tmp = (y / z) * x;
	} else if (z <= -26000000000000.0) {
		tmp = t * (x / z);
	} else if (z <= 1.15e+104) {
		tmp = y * (x / z);
	} else {
		tmp = x / (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.3e+177:
		tmp = x * (t / z)
	elif z <= -3.9e+55:
		tmp = (y / z) * x
	elif z <= -26000000000000.0:
		tmp = t * (x / z)
	elif z <= 1.15e+104:
		tmp = y * (x / z)
	else:
		tmp = x / (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.3e+177)
		tmp = Float64(x * Float64(t / z));
	elseif (z <= -3.9e+55)
		tmp = Float64(Float64(y / z) * x);
	elseif (z <= -26000000000000.0)
		tmp = Float64(t * Float64(x / z));
	elseif (z <= 1.15e+104)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(x / Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.3e+177)
		tmp = x * (t / z);
	elseif (z <= -3.9e+55)
		tmp = (y / z) * x;
	elseif (z <= -26000000000000.0)
		tmp = t * (x / z);
	elseif (z <= 1.15e+104)
		tmp = y * (x / z);
	else
		tmp = x / (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.3e+177], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.9e+55], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, -26000000000000.0], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.15e+104], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.2999999999999999e177

   1. Initial program 84.2%

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

   2. Taylor expanded in z around inf 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-/l* 82.2%

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

      3. associate-/r/ 82.4%

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

      4. cancel-sign-sub-inv 82.4%

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

      5. metadata-eval 82.4%

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

      6. *-lft-identity 82.4%

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

   4. Simplified 82.4%

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

   5. Taylor expanded in x around 0 78.8%

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

   6. Taylor expanded in y around 0 55.7%

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

      1. associate-/l* 55.3%

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

      2. associate-/r/ 62.9%

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

   8. Simplified 62.9%

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

   if -2.2999999999999999e177 < z < -3.90000000000000027e55

   1. Initial program 95.2%

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

   2. Taylor expanded in y around inf 60.2%

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

      1. associate-*l/ 68.8%

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

   4. Simplified 68.8%

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

   if -3.90000000000000027e55 < z < -2.6e13

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. neg-mul-1 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around 0 75.6%

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

      1. associate-/r/ 75.7%

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

   7. Applied egg-rr 75.7%

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

   if -2.6e13 < z < 1.14999999999999992e104

   1. Initial program 89.2%

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

      1. sub-neg 89.2%

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

      2. distribute-rgt-in 82.9%

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

      3. fma-def 85.0%

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

      4. distribute-neg-frac 85.0%

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

   3. Applied egg-rr 85.0%

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

   4. Taylor expanded in y around inf 70.4%

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

      1. associate-*r/ 71.7%

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

   6. Simplified 71.7%

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

   if 1.14999999999999992e104 < z

   1. Initial program 95.6%

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

   2. Taylor expanded in z around inf 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-/l* 95.7%

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

      3. neg-mul-1 95.7%

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

   4. Simplified 95.7%

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

   5. Taylor expanded in y around 0 69.5%

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{t}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+177}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq -3.9 \cdot 10^{+55}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \mathbf{elif}\;z \leq -26000000000000:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+104}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]

Alternative 10: 62.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+177}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+103}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.2e+177)
   (* x (/ t z))
   (if (<= z -5.2e+55)
     (/ x (/ z y))
     (if (<= z -4.8e+14)
       (* t (/ x z))
       (if (<= z 2.85e+103) (* y (/ x z)) (/ x (/ z t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.2e+177) {
		tmp = x * (t / z);
	} else if (z <= -5.2e+55) {
		tmp = x / (z / y);
	} else if (z <= -4.8e+14) {
		tmp = t * (x / z);
	} else if (z <= 2.85e+103) {
		tmp = y * (x / z);
	} else {
		tmp = x / (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 <= (-2.2d+177)) then
        tmp = x * (t / z)
    else if (z <= (-5.2d+55)) then
        tmp = x / (z / y)
    else if (z <= (-4.8d+14)) then
        tmp = t * (x / z)
    else if (z <= 2.85d+103) then
        tmp = y * (x / z)
    else
        tmp = x / (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 <= -2.2e+177) {
		tmp = x * (t / z);
	} else if (z <= -5.2e+55) {
		tmp = x / (z / y);
	} else if (z <= -4.8e+14) {
		tmp = t * (x / z);
	} else if (z <= 2.85e+103) {
		tmp = y * (x / z);
	} else {
		tmp = x / (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.2e+177:
		tmp = x * (t / z)
	elif z <= -5.2e+55:
		tmp = x / (z / y)
	elif z <= -4.8e+14:
		tmp = t * (x / z)
	elif z <= 2.85e+103:
		tmp = y * (x / z)
	else:
		tmp = x / (z / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.2e+177)
		tmp = Float64(x * Float64(t / z));
	elseif (z <= -5.2e+55)
		tmp = Float64(x / Float64(z / y));
	elseif (z <= -4.8e+14)
		tmp = Float64(t * Float64(x / z));
	elseif (z <= 2.85e+103)
		tmp = Float64(y * Float64(x / z));
	else
		tmp = Float64(x / Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.2e+177)
		tmp = x * (t / z);
	elseif (z <= -5.2e+55)
		tmp = x / (z / y);
	elseif (z <= -4.8e+14)
		tmp = t * (x / z);
	elseif (z <= 2.85e+103)
		tmp = y * (x / z);
	else
		tmp = x / (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.2e+177], N[(x * N[(t / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -5.2e+55], N[(x / N[(z / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -4.8e+14], N[(t * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.85e+103], N[(y * N[(x / z), $MachinePrecision]), $MachinePrecision], N[(x / N[(z / t), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -2.1999999999999998e177

   1. Initial program 84.2%

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

   2. Taylor expanded in z around inf 78.8%

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

      1. *-commutative 78.8%

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

      2. associate-/l* 82.2%

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

      3. associate-/r/ 82.4%

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

      4. cancel-sign-sub-inv 82.4%

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

      5. metadata-eval 82.4%

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

      6. *-lft-identity 82.4%

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

   4. Simplified 82.4%

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

   5. Taylor expanded in x around 0 78.8%

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

   6. Taylor expanded in y around 0 55.7%

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

      1. associate-/l* 55.3%

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

      2. associate-/r/ 62.9%

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

   8. Simplified 62.9%

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

   if -2.1999999999999998e177 < z < -5.2e55

   1. Initial program 95.2%

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

   2. Taylor expanded in z around inf 91.1%

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

      1. *-commutative 91.1%

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

      2. associate-/l* 95.5%

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

      3. neg-mul-1 95.5%

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

   4. Simplified 95.5%

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

   5. Taylor expanded in y around inf 69.1%

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

   if -5.2e55 < z < -4.8e14

   1. Initial program 99.7%

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

   2. Taylor expanded in z around inf 99.9%

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

      1. *-commutative 99.9%

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

      2. associate-/l* 99.9%

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

      3. neg-mul-1 99.9%

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

   4. Simplified 99.9%

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

   5. Taylor expanded in y around 0 75.6%

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

      1. associate-/r/ 75.7%

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

   7. Applied egg-rr 75.7%

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

   if -4.8e14 < z < 2.85000000000000016e103

   1. Initial program 89.2%

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

      1. sub-neg 89.2%

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

      2. distribute-rgt-in 82.9%

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

      3. fma-def 85.0%

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

      4. distribute-neg-frac 85.0%

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

   3. Applied egg-rr 85.0%

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

   4. Taylor expanded in y around inf 70.4%

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

      1. associate-*r/ 71.7%

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

   6. Simplified 71.7%

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

   if 2.85000000000000016e103 < z

   1. Initial program 95.6%

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

   2. Taylor expanded in z around inf 83.4%

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

      1. *-commutative 83.4%

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

      2. associate-/l* 95.7%

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

      3. neg-mul-1 95.7%

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

   4. Simplified 95.7%

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

   5. Taylor expanded in y around 0 69.5%

      \[\leadsto \frac{x}{\color{blue}{\frac{z}{t}}} \]
3. Recombined 5 regimes into one program.

4. Final simplification 70.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+177}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;z \leq -5.2 \cdot 10^{+55}:\\ \;\;\;\;\frac{x}{\frac{z}{y}}\\ \mathbf{elif}\;z \leq -4.8 \cdot 10^{+14}:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;z \leq 2.85 \cdot 10^{+103}:\\ \;\;\;\;y \cdot \frac{x}{z}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{\frac{z}{t}}\\ \end{array} \]

Alternative 11: 61.4% accurate, 2.2× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. sub-neg 90.8%

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

   2. distribute-rgt-in 87.2%

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

   3. fma-def 88.4%

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

   4. distribute-neg-frac 88.4%

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

3. Applied egg-rr 88.4%

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

4. Taylor expanded in y around inf 60.1%

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

   1. associate-*r/ 60.9%

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

6. Simplified 60.9%

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

7. Final simplification 60.9%

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

Developer target: 95.0% accurate, 0.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023216 
(FPCore (x y z t)
  :name "Numeric.SpecFunctions:invIncompleteBetaWorker from math-functions-0.1.5.2, C"
  :precision binary64

  :herbie-target
  (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)))))