System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2

Percentage Accurate: 60.2% → 98.1%

Time: 21.3s

Alternatives: 11

Speedup: 211.0×

• could not determine a ground truth

  1. x = 1.1026006573075033e+56
  2. y = 6.938311213590749e+218
  3. z = 6.739851152401732e+226
  4. t = 4.650160486770498e+214

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / 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 - (log(((1.0d0 - y) + (y * exp(z)))) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - (log(((1.0 - y) + (y * exp(z)))) / 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 - (log(((1.0d0 - y) + (y * exp(z)))) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (/ (log1p (* y (expm1 z))) t)))

C

double code(double x, double y, double z, double t) {
	return x - (log1p((y * expm1(z))) / t);
}

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log1p((y * Math.expm1(z))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log1p((y * math.expm1(z))) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log1p(Float64(y * expm1(z))) / t))
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

   1. associate-+l- 75.1%

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

   2. sub-neg 75.1%

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

   3. log1p-define 82.6%

      \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]

   4. neg-sub0 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]

   5. associate-+l- 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]

   6. neg-sub0 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]

   7. +-commutative 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(-y\right)}\right)}{t} \]

   8. unsub-neg 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} - y}\right)}{t} \]

   9. *-rgt-identity 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)}{t} \]

   10. distribute-lft-out-- 82.6%

       \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

   11. expm1-define 98.7%

       \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
4. Add Preprocessing

5. Final simplification 98.7%

   \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t} \]
6. Add Preprocessing

Alternative 2: 91.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.0002:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + 0.5 \cdot \left(y \cdot z\right)\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 0.0002)
   (- x (/ y (/ t (expm1 z))))
   (- x (/ (log1p (* z (+ y (* 0.5 (* y z))))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 0.0002) {
		tmp = x - (y / (t / expm1(z)));
	} else {
		tmp = x - (log1p((z * (y + (0.5 * (y * z))))) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.exp(z) <= 0.0002) {
		tmp = x - (y / (t / Math.expm1(z)));
	} else {
		tmp = x - (Math.log1p((z * (y + (0.5 * (y * z))))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.exp(z) <= 0.0002:
		tmp = x - (y / (t / math.expm1(z)))
	else:
		tmp = x - (math.log1p((z * (y + (0.5 * (y * z))))) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 0.0002)
		tmp = Float64(x - Float64(y / Float64(t / expm1(z))));
	else
		tmp = Float64(x - Float64(log1p(Float64(z * Float64(y + Float64(0.5 * Float64(y * z))))) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0002], N[(x - N[(y / N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(z * N[(y + N[(0.5 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 2.0000000000000001e-4

   1. Initial program 76.2%

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

   3. Taylor expanded in y around 0 77.5%

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

      1. associate-/l* 77.4%

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

      2. expm1-define 77.4%

         \[\leadsto x - y \cdot \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \]

   5. Simplified 77.4%

      \[\leadsto x - \color{blue}{y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}} \]
   6. Step-by-step derivation

      1. clear-num 77.4%

         \[\leadsto x - y \cdot \color{blue}{\frac{1}{\frac{t}{\mathsf{expm1}\left(z\right)}}} \]

      2. un-div-inv 77.5%

         \[\leadsto x - \color{blue}{\frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}} \]

   7. Applied egg-rr 77.5%

      \[\leadsto x - \color{blue}{\frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}} \]

   if 2.0000000000000001e-4 < (exp.f64 z)

   1. Initial program 48.5%

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

      1. associate-+l- 74.5%

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

      2. sub-neg 74.5%

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

      3. log1p-define 74.5%

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]

      4. neg-sub0 74.5%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]

      5. associate-+l- 74.5%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]

      6. neg-sub0 74.5%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]

      7. +-commutative 74.5%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(-y\right)}\right)}{t} \]

      8. unsub-neg 74.5%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} - y}\right)}{t} \]

      9. *-rgt-identity 74.5%

         \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)}{t} \]

      10. distribute-lft-out-- 74.5%

          \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

      11. expm1-define 98.1%

          \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   3. Simplified 98.1%

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
   4. Add Preprocessing

   5. Taylor expanded in z around 0 98.7%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{z \cdot \left(y + 0.5 \cdot \left(y \cdot z\right)\right)}\right)}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.0002:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + 0.5 \cdot \left(y \cdot z\right)\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+213} \lor \neg \left(y \leq 1.3 \cdot 10^{+113}\right):\\ \;\;\;\;x - \frac{\log \left(y \cdot z + 1\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.1e+213) (not (<= y 1.3e+113)))
   (- x (/ (log (+ (* y z) 1.0)) t))
   (- x (/ y (/ t (expm1 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.1e+213) || !(y <= 1.3e+113)) {
		tmp = x - (log(((y * z) + 1.0)) / t);
	} else {
		tmp = x - (y / (t / expm1(z)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.1e+213) || !(y <= 1.3e+113)) {
		tmp = x - (Math.log(((y * z) + 1.0)) / t);
	} else {
		tmp = x - (y / (t / Math.expm1(z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.1e+213) or not (y <= 1.3e+113):
		tmp = x - (math.log(((y * z) + 1.0)) / t)
	else:
		tmp = x - (y / (t / math.expm1(z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.1e+213) || !(y <= 1.3e+113))
		tmp = Float64(x - Float64(log(Float64(Float64(y * z) + 1.0)) / t));
	else
		tmp = Float64(x - Float64(y / Float64(t / expm1(z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.1e+213], N[Not[LessEqual[y, 1.3e+113]], $MachinePrecision]], N[(x - N[(N[Log[N[(N[(y * z), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.1000000000000001e213 or 1.3e113 < y

   1. Initial program 15.2%

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

   3. Taylor expanded in z around 0 81.3%

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

   if -2.1000000000000001e213 < y < 1.3e113

   1. Initial program 66.0%

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

   3. Taylor expanded in y around 0 80.9%

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

      1. associate-/l* 80.9%

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

      2. expm1-define 92.1%

         \[\leadsto x - y \cdot \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \]

   5. Simplified 92.1%

      \[\leadsto x - \color{blue}{y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}} \]
   6. Step-by-step derivation

      1. clear-num 92.1%

         \[\leadsto x - y \cdot \color{blue}{\frac{1}{\frac{t}{\mathsf{expm1}\left(z\right)}}} \]

      2. un-div-inv 92.1%

         \[\leadsto x - \color{blue}{\frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}} \]

   7. Applied egg-rr 92.1%

      \[\leadsto x - \color{blue}{\frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.1 \cdot 10^{+213} \lor \neg \left(y \leq 1.3 \cdot 10^{+113}\right):\\ \;\;\;\;x - \frac{\log \left(y \cdot z + 1\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{-1}{\frac{-0.5 \cdot \frac{z \cdot t}{y} + \frac{t}{y}}{z}}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+254}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{+198}:\\ \;\;\;\;y \cdot \frac{\mathsf{expm1}\left(z\right)}{-t}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-211}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \frac{-1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ -1.0 (/ (+ (* -0.5 (/ (* z t) y)) (/ t y)) z)))))
   (if (<= z -9.2e+254)
     t_1
     (if (<= z -3.7e+198)
       (* y (/ (expm1 z) (- t)))
       (if (<= z -5e-211) t_1 (+ x (* (* y z) (/ -1.0 t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z));
	double tmp;
	if (z <= -9.2e+254) {
		tmp = t_1;
	} else if (z <= -3.7e+198) {
		tmp = y * (expm1(z) / -t);
	} else if (z <= -5e-211) {
		tmp = t_1;
	} else {
		tmp = x + ((y * z) * (-1.0 / t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z));
	double tmp;
	if (z <= -9.2e+254) {
		tmp = t_1;
	} else if (z <= -3.7e+198) {
		tmp = y * (Math.expm1(z) / -t);
	} else if (z <= -5e-211) {
		tmp = t_1;
	} else {
		tmp = x + ((y * z) * (-1.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z))
	tmp = 0
	if z <= -9.2e+254:
		tmp = t_1
	elif z <= -3.7e+198:
		tmp = y * (math.expm1(z) / -t)
	elif z <= -5e-211:
		tmp = t_1
	else:
		tmp = x + ((y * z) * (-1.0 / t))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(-1.0 / Float64(Float64(Float64(-0.5 * Float64(Float64(z * t) / y)) + Float64(t / y)) / z)))
	tmp = 0.0
	if (z <= -9.2e+254)
		tmp = t_1;
	elseif (z <= -3.7e+198)
		tmp = Float64(y * Float64(expm1(z) / Float64(-t)));
	elseif (z <= -5e-211)
		tmp = t_1;
	else
		tmp = Float64(x + Float64(Float64(y * z) * Float64(-1.0 / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(-1.0 / N[(N[(N[(-0.5 * N[(N[(z * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e+254], t$95$1, If[LessEqual[z, -3.7e+198], N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -5e-211], t$95$1, N[(x + N[(N[(y * z), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.19999999999999994e254 or -3.6999999999999998e198 < z < -5.0000000000000002e-211

   1. Initial program 63.1%

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

   3. Taylor expanded in y around 0 74.8%

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

      1. associate-/l* 74.8%

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

      2. expm1-define 84.5%

         \[\leadsto x - y \cdot \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \]

   5. Simplified 84.5%

      \[\leadsto x - \color{blue}{y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 84.2%

         \[\leadsto x - \color{blue}{\frac{y \cdot \mathsf{expm1}\left(z\right)}{t}} \]

      2. clear-num 84.2%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)}}} \]

   7. Applied egg-rr 84.2%

      \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)}}} \]

   8. Taylor expanded in z around 0 76.0%

      \[\leadsto x - \frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{t \cdot z}{y} + \frac{t}{y}}{z}}} \]

   if -9.19999999999999994e254 < z < -3.6999999999999998e198

   1. Initial program 51.8%

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

   3. Taylor expanded in y around 0 66.6%

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

      1. associate-/l* 66.4%

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

      2. expm1-define 66.4%

         \[\leadsto x - y \cdot \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \]

   5. Simplified 66.4%

      \[\leadsto x - \color{blue}{y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}} \]

   6. Taylor expanded in x around 0 60.0%

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

      1. associate-*r/ 60.0%

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

      2. expm1-define 60.0%

         \[\leadsto \frac{-1 \cdot \left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

      3. neg-mul-1 60.0%

         \[\leadsto \frac{\color{blue}{-y \cdot \mathsf{expm1}\left(z\right)}}{t} \]

      4. distribute-rgt-neg-in 60.0%

         \[\leadsto \frac{\color{blue}{y \cdot \left(-\mathsf{expm1}\left(z\right)\right)}}{t} \]

      5. associate-/l* 59.8%

         \[\leadsto \color{blue}{y \cdot \frac{-\mathsf{expm1}\left(z\right)}{t}} \]

      6. distribute-neg-frac 59.8%

         \[\leadsto y \cdot \color{blue}{\left(-\frac{\mathsf{expm1}\left(z\right)}{t}\right)} \]

      7. distribute-neg-frac2 59.8%

         \[\leadsto y \cdot \color{blue}{\frac{\mathsf{expm1}\left(z\right)}{-t}} \]

   8. Simplified 59.8%

      \[\leadsto \color{blue}{y \cdot \frac{\mathsf{expm1}\left(z\right)}{-t}} \]

   if -5.0000000000000002e-211 < z

   1. Initial program 51.6%

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

      1. associate-+l- 77.2%

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

      2. sub-neg 77.2%

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

      3. log1p-define 77.2%

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]

      4. neg-sub0 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]

      5. associate-+l- 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]

      6. neg-sub0 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]

      7. +-commutative 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(-y\right)}\right)}{t} \]

      8. unsub-neg 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} - y}\right)}{t} \]

      9. *-rgt-identity 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)}{t} \]

      10. distribute-lft-out-- 77.2%

          \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

      11. expm1-define 97.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 97.6%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}}} \]

      2. associate-/r/ 97.6%

         \[\leadsto x - \color{blue}{\frac{1}{t} \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)} \]

   6. Applied egg-rr 97.6%

      \[\leadsto x - \color{blue}{\frac{1}{t} \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 97.3%

         \[\leadsto x - \frac{1}{t} \cdot \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)} \cdot \sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}\right) \cdot \sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}}\right) \]

      2. pow3 97.3%

         \[\leadsto x - \frac{1}{t} \cdot \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}\right)}^{3}}\right) \]

   8. Applied egg-rr 97.3%

      \[\leadsto x - \frac{1}{t} \cdot \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}\right)}^{3}}\right) \]

   9. Taylor expanded in z around 0 89.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+254}:\\ \;\;\;\;x + \frac{-1}{\frac{-0.5 \cdot \frac{z \cdot t}{y} + \frac{t}{y}}{z}}\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{+198}:\\ \;\;\;\;y \cdot \frac{\mathsf{expm1}\left(z\right)}{-t}\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-211}:\\ \;\;\;\;x + \frac{-1}{\frac{-0.5 \cdot \frac{z \cdot t}{y} + \frac{t}{y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \frac{-1}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{-1}{\frac{-0.5 \cdot \frac{z \cdot t}{y} + \frac{t}{y}}{z}}\\ \mathbf{if}\;z \leq -9.2 \cdot 10^{+254}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{+198}:\\ \;\;\;\;\mathsf{expm1}\left(z\right) \cdot \frac{y}{-t}\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-211}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \frac{-1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ x (/ -1.0 (/ (+ (* -0.5 (/ (* z t) y)) (/ t y)) z)))))
   (if (<= z -9.2e+254)
     t_1
     (if (<= z -3.7e+198)
       (* (expm1 z) (/ y (- t)))
       (if (<= z -4.7e-211) t_1 (+ x (* (* y z) (/ -1.0 t))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z));
	double tmp;
	if (z <= -9.2e+254) {
		tmp = t_1;
	} else if (z <= -3.7e+198) {
		tmp = expm1(z) * (y / -t);
	} else if (z <= -4.7e-211) {
		tmp = t_1;
	} else {
		tmp = x + ((y * z) * (-1.0 / t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z));
	double tmp;
	if (z <= -9.2e+254) {
		tmp = t_1;
	} else if (z <= -3.7e+198) {
		tmp = Math.expm1(z) * (y / -t);
	} else if (z <= -4.7e-211) {
		tmp = t_1;
	} else {
		tmp = x + ((y * z) * (-1.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z))
	tmp = 0
	if z <= -9.2e+254:
		tmp = t_1
	elif z <= -3.7e+198:
		tmp = math.expm1(z) * (y / -t)
	elif z <= -4.7e-211:
		tmp = t_1
	else:
		tmp = x + ((y * z) * (-1.0 / t))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x + Float64(-1.0 / Float64(Float64(Float64(-0.5 * Float64(Float64(z * t) / y)) + Float64(t / y)) / z)))
	tmp = 0.0
	if (z <= -9.2e+254)
		tmp = t_1;
	elseif (z <= -3.7e+198)
		tmp = Float64(expm1(z) * Float64(y / Float64(-t)));
	elseif (z <= -4.7e-211)
		tmp = t_1;
	else
		tmp = Float64(x + Float64(Float64(y * z) * Float64(-1.0 / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x + N[(-1.0 / N[(N[(N[(-0.5 * N[(N[(z * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -9.2e+254], t$95$1, If[LessEqual[z, -3.7e+198], N[(N[(Exp[z] - 1), $MachinePrecision] * N[(y / (-t)), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -4.7e-211], t$95$1, N[(x + N[(N[(y * z), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.19999999999999994e254 or -3.6999999999999998e198 < z < -4.6999999999999997e-211

   1. Initial program 63.1%

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

   3. Taylor expanded in y around 0 74.8%

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

      1. associate-/l* 74.8%

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

      2. expm1-define 84.5%

         \[\leadsto x - y \cdot \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \]

   5. Simplified 84.5%

      \[\leadsto x - \color{blue}{y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 84.2%

         \[\leadsto x - \color{blue}{\frac{y \cdot \mathsf{expm1}\left(z\right)}{t}} \]

      2. clear-num 84.2%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)}}} \]

   7. Applied egg-rr 84.2%

      \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)}}} \]

   8. Taylor expanded in z around 0 76.0%

      \[\leadsto x - \frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{t \cdot z}{y} + \frac{t}{y}}{z}}} \]

   if -9.19999999999999994e254 < z < -3.6999999999999998e198

   1. Initial program 51.8%

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

   3. Taylor expanded in y around 0 66.6%

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

      1. associate-/l* 66.4%

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

      2. expm1-define 66.4%

         \[\leadsto x - y \cdot \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \]

   5. Simplified 66.4%

      \[\leadsto x - \color{blue}{y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}} \]

   6. Taylor expanded in x around 0 60.0%

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

      1. associate-*r/ 60.0%

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

      2. expm1-define 60.0%

         \[\leadsto \frac{-1 \cdot \left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

      3. neg-mul-1 60.0%

         \[\leadsto \frac{\color{blue}{-y \cdot \mathsf{expm1}\left(z\right)}}{t} \]

      4. distribute-frac-neg 60.0%

         \[\leadsto \color{blue}{-\frac{y \cdot \mathsf{expm1}\left(z\right)}{t}} \]

      5. *-commutative 60.0%

         \[\leadsto -\frac{\color{blue}{\mathsf{expm1}\left(z\right) \cdot y}}{t} \]

      6. associate-*r/ 60.0%

         \[\leadsto -\color{blue}{\mathsf{expm1}\left(z\right) \cdot \frac{y}{t}} \]

      7. distribute-rgt-neg-in 60.0%

         \[\leadsto \color{blue}{\mathsf{expm1}\left(z\right) \cdot \left(-\frac{y}{t}\right)} \]

   8. Simplified 60.0%

      \[\leadsto \color{blue}{\mathsf{expm1}\left(z\right) \cdot \left(-\frac{y}{t}\right)} \]

   if -4.6999999999999997e-211 < z

   1. Initial program 51.6%

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

      1. associate-+l- 77.2%

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

      2. sub-neg 77.2%

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

      3. log1p-define 77.2%

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]

      4. neg-sub0 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]

      5. associate-+l- 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]

      6. neg-sub0 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]

      7. +-commutative 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(-y\right)}\right)}{t} \]

      8. unsub-neg 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} - y}\right)}{t} \]

      9. *-rgt-identity 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)}{t} \]

      10. distribute-lft-out-- 77.2%

          \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

      11. expm1-define 97.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 97.6%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}}} \]

      2. associate-/r/ 97.6%

         \[\leadsto x - \color{blue}{\frac{1}{t} \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)} \]

   6. Applied egg-rr 97.6%

      \[\leadsto x - \color{blue}{\frac{1}{t} \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 97.3%

         \[\leadsto x - \frac{1}{t} \cdot \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)} \cdot \sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}\right) \cdot \sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}}\right) \]

      2. pow3 97.3%

         \[\leadsto x - \frac{1}{t} \cdot \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}\right)}^{3}}\right) \]

   8. Applied egg-rr 97.3%

      \[\leadsto x - \frac{1}{t} \cdot \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}\right)}^{3}}\right) \]

   9. Taylor expanded in z around 0 89.3%

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

4. Final simplification 81.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.2 \cdot 10^{+254}:\\ \;\;\;\;x + \frac{-1}{\frac{-0.5 \cdot \frac{z \cdot t}{y} + \frac{t}{y}}{z}}\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{+198}:\\ \;\;\;\;\mathsf{expm1}\left(z\right) \cdot \frac{y}{-t}\\ \mathbf{elif}\;z \leq -4.7 \cdot 10^{-211}:\\ \;\;\;\;x + \frac{-1}{\frac{-0.5 \cdot \frac{z \cdot t}{y} + \frac{t}{y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \frac{-1}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 85.6% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (* y (/ (expm1 z) t))))

C

double code(double x, double y, double z, double t) {
	return x - (y * (expm1(z) / t));
}

Java

public static double code(double x, double y, double z, double t) {
	return x - (y * (Math.expm1(z) / t));
}

Python

def code(x, y, z, t):
	return x - (y * (math.expm1(z) / t))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(y * Float64(expm1(z) / t)))
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in y around 0 75.5%

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

   1. associate-/l* 75.4%

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

   2. expm1-define 85.6%

      \[\leadsto x - y \cdot \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \]

5. Simplified 85.6%

   \[\leadsto x - \color{blue}{y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}} \]

6. Final simplification 85.6%

   \[\leadsto x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t} \]
7. Add Preprocessing

Alternative 7: 85.7% accurate, 2.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (/ y (/ t (expm1 z)))))

C

double code(double x, double y, double z, double t) {
	return x - (y / (t / expm1(z)));
}

Java

public static double code(double x, double y, double z, double t) {
	return x - (y / (t / Math.expm1(z)));
}

Python

def code(x, y, z, t):
	return x - (y / (t / math.expm1(z)))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(y / Float64(t / expm1(z))))
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(y / N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 57.3%

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

3. Taylor expanded in y around 0 75.5%

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

   1. associate-/l* 75.4%

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

   2. expm1-define 85.6%

      \[\leadsto x - y \cdot \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \]

5. Simplified 85.6%

   \[\leadsto x - \color{blue}{y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}} \]
6. Step-by-step derivation

   1. clear-num 85.6%

      \[\leadsto x - y \cdot \color{blue}{\frac{1}{\frac{t}{\mathsf{expm1}\left(z\right)}}} \]

   2. un-div-inv 85.7%

      \[\leadsto x - \color{blue}{\frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}} \]

7. Applied egg-rr 85.7%

   \[\leadsto x - \color{blue}{\frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}} \]

8. Final simplification 85.7%

   \[\leadsto x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}} \]
9. Add Preprocessing

Alternative 8: 80.2% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-211}:\\ \;\;\;\;x + \frac{-1}{\frac{-0.5 \cdot \frac{z \cdot t}{y} + \frac{t}{y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \frac{-1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5e-211)
   (+ x (/ -1.0 (/ (+ (* -0.5 (/ (* z t) y)) (/ t y)) z)))
   (+ x (* (* y z) (/ -1.0 t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5e-211) {
		tmp = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z));
	} else {
		tmp = x + ((y * z) * (-1.0 / 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 <= (-5d-211)) then
        tmp = x + ((-1.0d0) / ((((-0.5d0) * ((z * t) / y)) + (t / y)) / z))
    else
        tmp = x + ((y * z) * ((-1.0d0) / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5e-211) {
		tmp = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z));
	} else {
		tmp = x + ((y * z) * (-1.0 / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5e-211:
		tmp = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z))
	else:
		tmp = x + ((y * z) * (-1.0 / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5e-211)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(Float64(-0.5 * Float64(Float64(z * t) / y)) + Float64(t / y)) / z)));
	else
		tmp = Float64(x + Float64(Float64(y * z) * Float64(-1.0 / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -5e-211)
		tmp = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z));
	else
		tmp = x + ((y * z) * (-1.0 / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5e-211], N[(x + N[(-1.0 / N[(N[(N[(-0.5 * N[(N[(z * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.0000000000000002e-211

   1. Initial program 62.0%

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

   3. Taylor expanded in y around 0 74.0%

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

      1. associate-/l* 73.9%

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

      2. expm1-define 82.7%

         \[\leadsto x - y \cdot \frac{\color{blue}{\mathsf{expm1}\left(z\right)}}{t} \]

   5. Simplified 82.7%

      \[\leadsto x - \color{blue}{y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}} \]
   6. Step-by-step derivation

      1. associate-*r/ 82.5%

         \[\leadsto x - \color{blue}{\frac{y \cdot \mathsf{expm1}\left(z\right)}{t}} \]

      2. clear-num 82.4%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)}}} \]

   7. Applied egg-rr 82.4%

      \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{y \cdot \mathsf{expm1}\left(z\right)}}} \]

   8. Taylor expanded in z around 0 70.8%

      \[\leadsto x - \frac{1}{\color{blue}{\frac{-0.5 \cdot \frac{t \cdot z}{y} + \frac{t}{y}}{z}}} \]

   if -5.0000000000000002e-211 < z

   1. Initial program 51.6%

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

      1. associate-+l- 77.2%

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

      2. sub-neg 77.2%

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

      3. log1p-define 77.2%

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]

      4. neg-sub0 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]

      5. associate-+l- 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]

      6. neg-sub0 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]

      7. +-commutative 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(-y\right)}\right)}{t} \]

      8. unsub-neg 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} - y}\right)}{t} \]

      9. *-rgt-identity 77.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)}{t} \]

      10. distribute-lft-out-- 77.2%

          \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

      11. expm1-define 97.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   3. Simplified 97.6%

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
   4. Add Preprocessing
   5. Step-by-step derivation

      1. clear-num 97.6%

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}}} \]

      2. associate-/r/ 97.6%

         \[\leadsto x - \color{blue}{\frac{1}{t} \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)} \]

   6. Applied egg-rr 97.6%

      \[\leadsto x - \color{blue}{\frac{1}{t} \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)} \]
   7. Step-by-step derivation

      1. add-cube-cbrt 97.3%

         \[\leadsto x - \frac{1}{t} \cdot \mathsf{log1p}\left(\color{blue}{\left(\sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)} \cdot \sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}\right) \cdot \sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}}\right) \]

      2. pow3 97.3%

         \[\leadsto x - \frac{1}{t} \cdot \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}\right)}^{3}}\right) \]

   8. Applied egg-rr 97.3%

      \[\leadsto x - \frac{1}{t} \cdot \mathsf{log1p}\left(\color{blue}{{\left(\sqrt[3]{y \cdot \mathsf{expm1}\left(z\right)}\right)}^{3}}\right) \]

   9. Taylor expanded in z around 0 89.3%

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

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{-211}:\\ \;\;\;\;x + \frac{-1}{\frac{-0.5 \cdot \frac{z \cdot t}{y} + \frac{t}{y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \left(y \cdot z\right) \cdot \frac{-1}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 70.9% accurate, 13.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-249}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-261}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -1.7e-249) x (if (<= t 6.4e-261) (* (/ z t) (- y)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.7e-249) {
		tmp = x;
	} else if (t <= 6.4e-261) {
		tmp = (z / t) * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= (-1.7d-249)) then
        tmp = x
    else if (t <= 6.4d-261) then
        tmp = (z / t) * -y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -1.7e-249) {
		tmp = x;
	} else if (t <= 6.4e-261) {
		tmp = (z / t) * -y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -1.7e-249:
		tmp = x
	elif t <= 6.4e-261:
		tmp = (z / t) * -y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -1.7e-249)
		tmp = x;
	elseif (t <= 6.4e-261)
		tmp = Float64(Float64(z / t) * Float64(-y));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -1.7e-249)
		tmp = x;
	elseif (t <= 6.4e-261)
		tmp = (z / t) * -y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -1.7e-249], x, If[LessEqual[t, 6.4e-261], N[(N[(z / t), $MachinePrecision] * (-y)), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -1.6999999999999999e-249 or 6.40000000000000008e-261 < t

   1. Initial program 59.7%

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

      1. associate-+l- 78.9%

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

      2. sub-neg 78.9%

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

      3. log1p-define 86.1%

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]

      4. neg-sub0 86.1%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]

      5. associate-+l- 86.1%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]

      6. neg-sub0 86.1%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]

      7. +-commutative 86.1%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(-y\right)}\right)}{t} \]

      8. unsub-neg 86.1%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} - y}\right)}{t} \]

      9. *-rgt-identity 86.1%

         \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)}{t} \]

      10. distribute-lft-out-- 86.1%

          \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

      11. expm1-define 98.9%

          \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   3. Simplified 98.9%

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 73.2%

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

   if -1.6999999999999999e-249 < t < 6.40000000000000008e-261

   1. Initial program 24.5%

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

      1. associate-+l- 24.3%

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

      2. sub-neg 24.3%

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

      3. log1p-define 35.2%

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]

      4. neg-sub0 35.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]

      5. associate-+l- 35.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]

      6. neg-sub0 35.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]

      7. +-commutative 35.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(-y\right)}\right)}{t} \]

      8. unsub-neg 35.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} - y}\right)}{t} \]

      9. *-rgt-identity 35.2%

         \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)}{t} \]

      10. distribute-lft-out-- 35.2%

          \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

      11. expm1-define 95.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   3. Simplified 95.6%

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around 0 18.7%

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

      1. mul-1-neg 18.7%

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

      2. log1p-define 29.5%

         \[\leadsto -\frac{\color{blue}{\mathsf{log1p}\left(y \cdot \left(e^{z} - 1\right)\right)}}{t} \]

      3. expm1-define 90.2%

         \[\leadsto -\frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

      4. distribute-frac-neg2 90.2%

         \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}} \]

   7. Simplified 90.2%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}} \]

   8. Taylor expanded in z around 0 59.4%

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

      1. mul-1-neg 59.4%

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

      2. associate-/l* 63.5%

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

      3. distribute-rgt-neg-in 63.5%

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

   10. Simplified 63.5%

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

4. Final simplification 72.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7 \cdot 10^{-249}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 6.4 \cdot 10^{-261}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-y\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 80.9% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+86}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.85e+86) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.85e+86) {
		tmp = x;
	} 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 <= (-2.85d+86)) then
        tmp = x
    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 <= -2.85e+86) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.85e+86:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.85e+86)
		tmp = x;
	else
		tmp = Float64(x - Float64(y * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.85e+86)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.85e+86], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.85e86

   1. Initial program 72.2%

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

      1. associate-+l- 72.2%

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

      2. sub-neg 72.2%

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

      3. log1p-define 99.9%

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]

      4. neg-sub0 99.9%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]

      5. associate-+l- 99.9%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]

      6. neg-sub0 99.9%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]

      7. +-commutative 99.9%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(-y\right)}\right)}{t} \]

      8. unsub-neg 99.9%

         \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} - y}\right)}{t} \]

      9. *-rgt-identity 99.9%

         \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)}{t} \]

      10. distribute-lft-out-- 99.9%

          \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

      11. expm1-define 99.9%

          \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
   4. Add Preprocessing

   5. Taylor expanded in x around inf 49.4%

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

   if -2.85e86 < z

   1. Initial program 53.5%

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

   3. Taylor expanded in z around 0 85.8%

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

      1. associate-/l* 85.9%

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

   5. Simplified 85.9%

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

4. Final simplification 78.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.85 \cdot 10^{+86}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.5% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

function code(x, y, z, t)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 57.3%

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

   1. associate-+l- 75.1%

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

   2. sub-neg 75.1%

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

   3. log1p-define 82.6%

      \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(-\left(y - y \cdot e^{z}\right)\right)}}{t} \]

   4. neg-sub0 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{0 - \left(y - y \cdot e^{z}\right)}\right)}{t} \]

   5. associate-+l- 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(0 - y\right) + y \cdot e^{z}}\right)}{t} \]

   6. neg-sub0 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{\left(-y\right)} + y \cdot e^{z}\right)}{t} \]

   7. +-commutative 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} + \left(-y\right)}\right)}{t} \]

   8. unsub-neg 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot e^{z} - y}\right)}{t} \]

   9. *-rgt-identity 82.6%

      \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot e^{z} - \color{blue}{y \cdot 1}\right)}{t} \]

   10. distribute-lft-out-- 82.6%

       \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{y \cdot \left(e^{z} - 1\right)}\right)}{t} \]

   11. expm1-define 98.7%

       \[\leadsto x - \frac{\mathsf{log1p}\left(y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}\right)}{t} \]

3. Simplified 98.7%

   \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
4. Add Preprocessing

5. Taylor expanded in x around inf 68.6%

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

6. Final simplification 68.6%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 73.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5}{y \cdot t}\\ \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{t\_1}{z \cdot z}\right) - t\_1 \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- 0.5) (* y t))))
   (if (< z -2.8874623088207947e+119)
     (- (- x (/ t_1 (* z z))) (* t_1 (/ (/ 2.0 z) (* z z))))
     (- x (/ (log (+ 1.0 (* z y))) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (log((1.0 + (z * y))) / 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) :: tmp
    t_1 = -0.5d0 / (y * t)
    if (z < (-2.8874623088207947d+119)) then
        tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0d0 / z) / (z * z)))
    else
        tmp = x - (log((1.0d0 + (z * y))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (Math.log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -0.5 / (y * t)
	tmp = 0
	if z < -2.8874623088207947e+119:
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)))
	else:
		tmp = x - (math.log((1.0 + (z * y))) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5) / Float64(y * t))
	tmp = 0.0
	if (z < -2.8874623088207947e+119)
		tmp = Float64(Float64(x - Float64(t_1 / Float64(z * z))) - Float64(t_1 * Float64(Float64(2.0 / z) / Float64(z * z))));
	else
		tmp = Float64(x - Float64(log(Float64(1.0 + Float64(z * y))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -0.5 / (y * t);
	tmp = 0.0;
	if (z < -2.8874623088207947e+119)
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	else
		tmp = x - (log((1.0 + (z * y))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-0.5) / N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.8874623088207947e+119], N[(N[(x - N[(t$95$1 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[(2.0 / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(1.0 + N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024078 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (if (< z -2.8874623088207947e+119) (- (- x (/ (/ (- 0.5) (* y t)) (* z z))) (* (/ (- 0.5) (* y t)) (/ (/ 2.0 z) (* z z)))) (- x (/ (log (+ 1.0 (* z y))) t)))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))