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

Percentage Accurate: 62.2% → 98.3%

Time: 22.2s

Alternatives: 12

Speedup: 211.0×

• could not determine a ground truth

  1. x = -6.43777217177086e-28
  2. y = 5.739477219846883e+78
  3. z = 1.8294800714365586e+132
  4. t = -5.025144829588348e+295

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: 62.2% 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: 98.3% 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 61.1%

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

   1. associate-+l- 74.1%

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

   2. sub-neg 74.1%

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

   3. log1p-define 80.9%

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

   4. neg-sub0 80.9%

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

   5. associate-+l- 80.9%

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

   6. neg-sub0 80.9%

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

   7. +-commutative 80.9%

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

   8. unsub-neg 80.9%

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

   9. *-rgt-identity 80.9%

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

   10. distribute-lft-out-- 80.9%

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

   11. expm1-define 97.7%

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

3. Simplified 97.7%

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

Alternative 2: 91.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+61}:\\ \;\;\;\;x + \frac{-1}{\frac{0.5 \cdot \left(y \cdot t\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)}}{y}}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-14}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + 0.5 \cdot \left(y \cdot z\right)\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8.6e+61)
   (+
    x
    (/
     -1.0
     (/
      (+
       (* 0.5 (* y t))
       (/ t (* z (+ 1.0 (* z (+ 0.5 (* z 0.16666666666666666)))))))
      y)))
   (if (<= y 2.15e-14)
     (- 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 (y <= -8.6e+61) {
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	} else if (y <= 2.15e-14) {
		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 (y <= -8.6e+61) {
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	} else if (y <= 2.15e-14) {
		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 y <= -8.6e+61:
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y))
	elif y <= 2.15e-14:
		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 (y <= -8.6e+61)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(Float64(0.5 * Float64(y * t)) + Float64(t / Float64(z * Float64(1.0 + Float64(z * Float64(0.5 + Float64(z * 0.16666666666666666))))))) / y)));
	elseif (y <= 2.15e-14)
		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[y, -8.6e+61], N[(x + N[(-1.0 / N[(N[(N[(0.5 * N[(y * t), $MachinePrecision]), $MachinePrecision] + N[(t / N[(z * N[(1.0 + N[(z * N[(0.5 + N[(z * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 2.15e-14], 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 3 regimes

2. if y < -8.6000000000000003e61

   1. Initial program 54.3%

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

      1. associate-+l- 82.1%

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

      2. sub-neg 82.1%

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

      3. log1p-define 82.1%

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

      4. neg-sub0 82.1%

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

      5. associate-+l- 82.1%

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

      6. neg-sub0 82.1%

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

      7. +-commutative 82.1%

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

      8. unsub-neg 82.1%

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

      9. *-rgt-identity 82.1%

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

      10. distribute-lft-out-- 82.1%

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

      11. expm1-define 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. clear-num 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in z around 0 55.7%

      \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(\color{blue}{z \cdot \left(y + z \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) + 0.5 \cdot y\right)\right)}\right)}} \]

   10. Taylor expanded in y around 0 77.6%

       \[\leadsto x - \frac{1}{\color{blue}{\frac{0.5 \cdot \left(t \cdot y\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + 0.16666666666666666 \cdot z\right)\right)}}{y}}} \]

   if -8.6000000000000003e61 < y < 2.14999999999999999e-14

   1. Initial program 75.9%

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

      1. associate-+l- 77.1%

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

      2. sub-neg 77.1%

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

      3. log1p-define 87.5%

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

      4. neg-sub0 87.5%

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

      5. associate-+l- 87.5%

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

      6. neg-sub0 87.5%

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

      7. +-commutative 87.5%

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

      8. unsub-neg 87.5%

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

      9. *-rgt-identity 87.5%

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

      10. distribute-lft-out-- 87.5%

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

      11. expm1-define 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in y around 0 84.1%

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

      1. associate-/l* 84.1%

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

      2. expm1-define 95.7%

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

   7. Simplified 95.7%

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

      1. clear-num 95.7%

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

      2. un-div-inv 95.8%

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

   9. Applied egg-rr 95.8%

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

   if 2.14999999999999999e-14 < y

   1. Initial program 11.1%

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

      1. associate-+l- 54.5%

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

      2. sub-neg 54.5%

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

      3. log1p-define 54.5%

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

      4. neg-sub0 54.5%

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

      5. associate-+l- 54.5%

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

      6. neg-sub0 54.5%

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

      7. +-commutative 54.5%

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

      8. unsub-neg 54.5%

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

      9. *-rgt-identity 54.5%

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

      10. distribute-lft-out-- 54.5%

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

      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 z around 0 98.1%

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

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.6 \cdot 10^{+61}:\\ \;\;\;\;x + \frac{-1}{\frac{0.5 \cdot \left(y \cdot t\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)}}{y}}\\ \mathbf{elif}\;y \leq 2.15 \cdot 10^{-14}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + 0.5 \cdot \left(y \cdot z\right)\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+61}:\\ \;\;\;\;x + \frac{-1}{\frac{0.5 \cdot \left(y \cdot t\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)}}{y}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+252}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(z \cdot \left(y + 0.5 \cdot \left(y \cdot z\right)\right)\right)}{-t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.5e+61)
   (+
    x
    (/
     -1.0
     (/
      (+
       (* 0.5 (* y t))
       (/ t (* z (+ 1.0 (* z (+ 0.5 (* z 0.16666666666666666)))))))
      y)))
   (if (<= y 1.1e+252)
     (- x (/ y (/ t (expm1 z))))
     (/ (log1p (* z (+ y (* 0.5 (* y z))))) (- t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.5e+61) {
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	} else if (y <= 1.1e+252) {
		tmp = x - (y / (t / expm1(z)));
	} else {
		tmp = 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 (y <= -9.5e+61) {
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	} else if (y <= 1.1e+252) {
		tmp = x - (y / (t / Math.expm1(z)));
	} else {
		tmp = Math.log1p((z * (y + (0.5 * (y * z))))) / -t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -9.5e+61:
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y))
	elif y <= 1.1e+252:
		tmp = x - (y / (t / math.expm1(z)))
	else:
		tmp = math.log1p((z * (y + (0.5 * (y * z))))) / -t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.5e+61)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(Float64(0.5 * Float64(y * t)) + Float64(t / Float64(z * Float64(1.0 + Float64(z * Float64(0.5 + Float64(z * 0.16666666666666666))))))) / y)));
	elseif (y <= 1.1e+252)
		tmp = Float64(x - Float64(y / Float64(t / expm1(z))));
	else
		tmp = Float64(log1p(Float64(z * Float64(y + Float64(0.5 * Float64(y * z))))) / Float64(-t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9.5e+61], N[(x + N[(-1.0 / N[(N[(N[(0.5 * N[(y * t), $MachinePrecision]), $MachinePrecision] + N[(t / N[(z * N[(1.0 + N[(z * N[(0.5 + N[(z * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.1e+252], N[(x - N[(y / N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[Log[1 + N[(z * N[(y + N[(0.5 * N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.49999999999999959e61

   1. Initial program 54.3%

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

      1. associate-+l- 82.1%

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

      2. sub-neg 82.1%

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

      3. log1p-define 82.1%

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

      4. neg-sub0 82.1%

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

      5. associate-+l- 82.1%

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

      6. neg-sub0 82.1%

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

      7. +-commutative 82.1%

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

      8. unsub-neg 82.1%

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

      9. *-rgt-identity 82.1%

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

      10. distribute-lft-out-- 82.1%

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

      11. expm1-define 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. clear-num 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in z around 0 55.7%

      \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(\color{blue}{z \cdot \left(y + z \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) + 0.5 \cdot y\right)\right)}\right)}} \]

   10. Taylor expanded in y around 0 77.6%

       \[\leadsto x - \frac{1}{\color{blue}{\frac{0.5 \cdot \left(t \cdot y\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + 0.16666666666666666 \cdot z\right)\right)}}{y}}} \]

   if -9.49999999999999959e61 < y < 1.1e252

   1. Initial program 64.3%

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

      1. associate-+l- 74.4%

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

      2. sub-neg 74.4%

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

      3. log1p-define 82.9%

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

      4. neg-sub0 82.9%

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

      5. associate-+l- 82.9%

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

      6. neg-sub0 82.9%

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

      7. +-commutative 82.9%

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

      8. unsub-neg 82.9%

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

      9. *-rgt-identity 82.9%

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

      10. distribute-lft-out-- 82.9%

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

      11. expm1-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in y around 0 79.3%

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

      1. associate-/l* 79.3%

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

      2. expm1-define 93.7%

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

   7. Simplified 93.7%

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

      1. clear-num 93.6%

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

      2. un-div-inv 93.7%

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

   9. Applied egg-rr 93.7%

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

   if 1.1e252 < y

   1. Initial program 0.3%

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

      1. associate-+l- 3.7%

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

      2. sub-neg 3.7%

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

      3. log1p-define 3.7%

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

      4. neg-sub0 3.7%

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

      5. associate-+l- 3.7%

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

      6. neg-sub0 3.7%

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

      7. +-commutative 3.7%

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

      8. unsub-neg 3.7%

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

      9. *-rgt-identity 3.7%

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

      10. distribute-lft-out-- 3.7%

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

      11. expm1-define 99.5%

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

   3. Simplified 99.5%

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

   5. Taylor expanded in x around 0 2.3%

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

      1. mul-1-neg 2.3%

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

      2. log1p-define 2.3%

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

      3. expm1-define 99.2%

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

      4. distribute-frac-neg2 99.2%

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

   7. Simplified 99.2%

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

   8. Taylor expanded in z around 0 99.2%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+61}:\\ \;\;\;\;x + \frac{-1}{\frac{0.5 \cdot \left(y \cdot t\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)}}{y}}\\ \mathbf{elif}\;y \leq 1.1 \cdot 10^{+252}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(z \cdot \left(y + 0.5 \cdot \left(y \cdot z\right)\right)\right)}{-t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 93.6% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -75000000:\\ \;\;\;\;x + \frac{1}{\frac{\frac{t}{1 - e^{z}} - 0.5 \cdot \left(y \cdot t\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + y \cdot \left(z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -75000000.0)
   (+ x (/ 1.0 (/ (- (/ t (- 1.0 (exp z))) (* 0.5 (* y t))) y)))
   (-
    x
    (/ (log1p (* z (+ y (* y (* z (+ 0.5 (* z 0.16666666666666666))))))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -75000000.0) {
		tmp = x + (1.0 / (((t / (1.0 - exp(z))) - (0.5 * (y * t))) / y));
	} else {
		tmp = x - (log1p((z * (y + (y * (z * (0.5 + (z * 0.16666666666666666))))))) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -75000000.0) {
		tmp = x + (1.0 / (((t / (1.0 - Math.exp(z))) - (0.5 * (y * t))) / y));
	} else {
		tmp = x - (Math.log1p((z * (y + (y * (z * (0.5 + (z * 0.16666666666666666))))))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -75000000.0:
		tmp = x + (1.0 / (((t / (1.0 - math.exp(z))) - (0.5 * (y * t))) / y))
	else:
		tmp = x - (math.log1p((z * (y + (y * (z * (0.5 + (z * 0.16666666666666666))))))) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -75000000.0)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(Float64(t / Float64(1.0 - exp(z))) - Float64(0.5 * Float64(y * t))) / y)));
	else
		tmp = Float64(x - Float64(log1p(Float64(z * Float64(y + Float64(y * Float64(z * Float64(0.5 + Float64(z * 0.16666666666666666))))))) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -75000000.0], N[(x + N[(1.0 / N[(N[(N[(t / N[(1.0 - N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(z * N[(y + N[(y * N[(z * N[(0.5 + N[(z * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.5e7

   1. Initial program 77.6%

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

      1. associate-+l- 77.6%

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

      2. sub-neg 77.6%

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

      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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. clear-num 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 83.7%

      \[\leadsto x - \frac{1}{\color{blue}{\frac{0.5 \cdot \left(t \cdot y\right) + \frac{t}{e^{z} - 1}}{y}}} \]

   if -7.5e7 < z

   1. Initial program 53.7%

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

      1. associate-+l- 72.5%

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

      2. sub-neg 72.5%

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

      3. log1p-define 72.5%

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

      4. neg-sub0 72.5%

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

      5. associate-+l- 72.5%

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

      6. neg-sub0 72.5%

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

      7. +-commutative 72.5%

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

      8. unsub-neg 72.5%

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

      9. *-rgt-identity 72.5%

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

      10. distribute-lft-out-- 72.5%

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

      11. expm1-define 96.8%

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

   3. Simplified 96.8%

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

      1. clear-num 96.8%

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

      2. associate-/r/ 96.8%

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

   6. Applied egg-rr 96.8%

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

      1. associate-*l/ 96.8%

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

      2. *-un-lft-identity 96.8%

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

      3. clear-num 96.8%

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

   8. Applied egg-rr 96.8%

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

   9. Taylor expanded in z around 0 96.5%

      \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(\color{blue}{z \cdot \left(y + z \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) + 0.5 \cdot y\right)\right)}\right)}} \]

   10. Taylor expanded in t around 0 78.4%

       \[\leadsto x - \color{blue}{\frac{\log \left(1 + z \cdot \left(y + z \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) + 0.5 \cdot y\right)\right)\right)}{t}} \]
   11. Step-by-step derivation

       1. log1p-define 96.6%

          \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(z \cdot \left(y + z \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) + 0.5 \cdot y\right)\right)\right)}}{t} \]

       2. distribute-rgt-in 96.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + \color{blue}{\left(\left(0.16666666666666666 \cdot \left(y \cdot z\right)\right) \cdot z + \left(0.5 \cdot y\right) \cdot z\right)}\right)\right)}{t} \]

       3. *-commutative 96.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + \left(\color{blue}{\left(\left(y \cdot z\right) \cdot 0.16666666666666666\right)} \cdot z + \left(0.5 \cdot y\right) \cdot z\right)\right)\right)}{t} \]

       4. associate-*r* 96.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + \left(\left(\left(y \cdot z\right) \cdot 0.16666666666666666\right) \cdot z + \color{blue}{0.5 \cdot \left(y \cdot z\right)}\right)\right)\right)}{t} \]

       5. *-commutative 96.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + \left(\left(\left(y \cdot z\right) \cdot 0.16666666666666666\right) \cdot z + \color{blue}{\left(y \cdot z\right) \cdot 0.5}\right)\right)\right)}{t} \]

       6. +-commutative 96.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + \color{blue}{\left(\left(y \cdot z\right) \cdot 0.5 + \left(\left(y \cdot z\right) \cdot 0.16666666666666666\right) \cdot z\right)}\right)\right)}{t} \]

       7. associate-*l* 96.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + \left(\left(y \cdot z\right) \cdot 0.5 + \color{blue}{\left(y \cdot z\right) \cdot \left(0.16666666666666666 \cdot z\right)}\right)\right)\right)}{t} \]

       8. distribute-lft-in 96.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + \color{blue}{\left(y \cdot z\right) \cdot \left(0.5 + 0.16666666666666666 \cdot z\right)}\right)\right)}{t} \]

       9. associate-*r* 96.6%

          \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + \color{blue}{y \cdot \left(z \cdot \left(0.5 + 0.16666666666666666 \cdot z\right)\right)}\right)\right)}{t} \]

       10. *-commutative 96.6%

           \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + y \cdot \left(z \cdot \left(0.5 + \color{blue}{z \cdot 0.16666666666666666}\right)\right)\right)\right)}{t} \]

   12. Simplified 96.6%

       \[\leadsto x - \color{blue}{\frac{\mathsf{log1p}\left(z \cdot \left(y + y \cdot \left(z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)\right)\right)}{t}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -75000000:\\ \;\;\;\;x + \frac{1}{\frac{\frac{t}{1 - e^{z}} - 0.5 \cdot \left(y \cdot t\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + y \cdot \left(z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 93.8% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.13:\\ \;\;\;\;x + \frac{1}{\frac{\frac{t}{1 - e^{z}} - 0.5 \cdot \left(y \cdot t\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + 0.5 \cdot \left(y \cdot z\right)\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.13)
   (+ x (/ 1.0 (/ (- (/ t (- 1.0 (exp z))) (* 0.5 (* y t))) y)))
   (- x (/ (log1p (* z (+ y (* 0.5 (* y z))))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.13) {
		tmp = x + (1.0 / (((t / (1.0 - exp(z))) - (0.5 * (y * t))) / y));
	} 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 (z <= -0.13) {
		tmp = x + (1.0 / (((t / (1.0 - Math.exp(z))) - (0.5 * (y * t))) / y));
	} else {
		tmp = x - (Math.log1p((z * (y + (0.5 * (y * z))))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -0.13:
		tmp = x + (1.0 / (((t / (1.0 - math.exp(z))) - (0.5 * (y * t))) / y))
	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 (z <= -0.13)
		tmp = Float64(x + Float64(1.0 / Float64(Float64(Float64(t / Float64(1.0 - exp(z))) - Float64(0.5 * Float64(y * t))) / y)));
	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[z, -0.13], N[(x + N[(1.0 / N[(N[(N[(t / N[(1.0 - N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(y * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $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 z < -0.13

   1. Initial program 78.7%

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

      1. associate-+l- 78.7%

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

      2. sub-neg 78.7%

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

      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. Step-by-step derivation

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

      2. *-un-lft-identity 99.9%

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

      3. clear-num 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in y around 0 83.4%

      \[\leadsto x - \frac{1}{\color{blue}{\frac{0.5 \cdot \left(t \cdot y\right) + \frac{t}{e^{z} - 1}}{y}}} \]

   if -0.13 < z

   1. Initial program 52.7%

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

      1. associate-+l- 71.9%

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

      2. sub-neg 71.9%

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

      3. log1p-define 71.9%

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

      4. neg-sub0 71.9%

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

      5. associate-+l- 71.9%

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

      6. neg-sub0 71.9%

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

      7. +-commutative 71.9%

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

      8. unsub-neg 71.9%

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

      9. *-rgt-identity 71.9%

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

      10. distribute-lft-out-- 71.9%

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

      11. expm1-define 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in z around 0 96.8%

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

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.13:\\ \;\;\;\;x + \frac{1}{\frac{\frac{t}{1 - e^{z}} - 0.5 \cdot \left(y \cdot t\right)}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + 0.5 \cdot \left(y \cdot z\right)\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+61}:\\ \;\;\;\;x + \frac{-1}{\frac{0.5 \cdot \left(y \cdot t\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -8.5e+61)
   (+
    x
    (/
     -1.0
     (/
      (+
       (* 0.5 (* y t))
       (/ t (* z (+ 1.0 (* z (+ 0.5 (* z 0.16666666666666666)))))))
      y)))
   (- x (/ y (/ t (expm1 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -8.5e+61) {
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	} 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 <= -8.5e+61) {
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	} else {
		tmp = x - (y / (t / Math.expm1(z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -8.5e+61:
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y))
	else:
		tmp = x - (y / (t / math.expm1(z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -8.5e+61)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(Float64(0.5 * Float64(y * t)) + Float64(t / Float64(z * Float64(1.0 + Float64(z * Float64(0.5 + Float64(z * 0.16666666666666666))))))) / y)));
	else
		tmp = Float64(x - Float64(y / Float64(t / expm1(z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -8.5e+61], N[(x + N[(-1.0 / N[(N[(N[(0.5 * N[(y * t), $MachinePrecision]), $MachinePrecision] + N[(t / N[(z * N[(1.0 + N[(z * N[(0.5 + N[(z * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 < -8.50000000000000035e61

   1. Initial program 54.3%

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

      1. associate-+l- 82.1%

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

      2. sub-neg 82.1%

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

      3. log1p-define 82.1%

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

      4. neg-sub0 82.1%

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

      5. associate-+l- 82.1%

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

      6. neg-sub0 82.1%

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

      7. +-commutative 82.1%

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

      8. unsub-neg 82.1%

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

      9. *-rgt-identity 82.1%

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

      10. distribute-lft-out-- 82.1%

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

      11. expm1-define 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. clear-num 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in z around 0 55.7%

      \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(\color{blue}{z \cdot \left(y + z \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) + 0.5 \cdot y\right)\right)}\right)}} \]

   10. Taylor expanded in y around 0 77.6%

       \[\leadsto x - \frac{1}{\color{blue}{\frac{0.5 \cdot \left(t \cdot y\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + 0.16666666666666666 \cdot z\right)\right)}}{y}}} \]

   if -8.50000000000000035e61 < y

   1. Initial program 62.5%

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

      1. associate-+l- 72.4%

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

      2. sub-neg 72.4%

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

      3. log1p-define 80.7%

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

      4. neg-sub0 80.7%

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

      5. associate-+l- 80.7%

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

      6. neg-sub0 80.7%

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

      7. +-commutative 80.7%

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

      8. unsub-neg 80.7%

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

      9. *-rgt-identity 80.7%

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

      10. distribute-lft-out-- 80.7%

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

      11. expm1-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in y around 0 77.2%

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

      1. associate-/l* 77.1%

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

      2. expm1-define 91.2%

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

   7. Simplified 91.2%

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

      1. clear-num 91.2%

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

      2. un-div-inv 91.2%

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

   9. Applied egg-rr 91.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+61}:\\ \;\;\;\;x + \frac{-1}{\frac{0.5 \cdot \left(y \cdot t\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 88.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+62}:\\ \;\;\;\;x + \frac{-1}{\frac{0.5 \cdot \left(y \cdot t\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.65e+62)
   (+
    x
    (/
     -1.0
     (/
      (+
       (* 0.5 (* y t))
       (/ t (* z (+ 1.0 (* z (+ 0.5 (* z 0.16666666666666666)))))))
      y)))
   (- x (* y (/ (expm1 z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.65e+62) {
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	} else {
		tmp = x - (y * (expm1(z) / t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.65e+62) {
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	} else {
		tmp = x - (y * (Math.expm1(z) / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.65e+62:
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y))
	else:
		tmp = x - (y * (math.expm1(z) / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.65e+62)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(Float64(0.5 * Float64(y * t)) + Float64(t / Float64(z * Float64(1.0 + Float64(z * Float64(0.5 + Float64(z * 0.16666666666666666))))))) / y)));
	else
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.65e+62], N[(x + N[(-1.0 / N[(N[(N[(0.5 * N[(y * t), $MachinePrecision]), $MachinePrecision] + N[(t / N[(z * N[(1.0 + N[(z * N[(0.5 + N[(z * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.65e62

   1. Initial program 54.3%

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

      1. associate-+l- 82.1%

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

      2. sub-neg 82.1%

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

      3. log1p-define 82.1%

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

      4. neg-sub0 82.1%

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

      5. associate-+l- 82.1%

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

      6. neg-sub0 82.1%

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

      7. +-commutative 82.1%

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

      8. unsub-neg 82.1%

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

      9. *-rgt-identity 82.1%

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

      10. distribute-lft-out-- 82.1%

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

      11. expm1-define 99.7%

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

   3. Simplified 99.7%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.7%

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

      2. *-un-lft-identity 99.7%

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

      3. clear-num 99.7%

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

   8. Applied egg-rr 99.7%

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

   9. Taylor expanded in z around 0 55.7%

      \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(\color{blue}{z \cdot \left(y + z \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) + 0.5 \cdot y\right)\right)}\right)}} \]

   10. Taylor expanded in y around 0 77.6%

       \[\leadsto x - \frac{1}{\color{blue}{\frac{0.5 \cdot \left(t \cdot y\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + 0.16666666666666666 \cdot z\right)\right)}}{y}}} \]

   if -1.65e62 < y

   1. Initial program 62.5%

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

      1. associate-+l- 72.4%

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

      2. sub-neg 72.4%

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

      3. log1p-define 80.7%

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

      4. neg-sub0 80.7%

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

      5. associate-+l- 80.7%

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

      6. neg-sub0 80.7%

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

      7. +-commutative 80.7%

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

      8. unsub-neg 80.7%

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

      9. *-rgt-identity 80.7%

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

      10. distribute-lft-out-- 80.7%

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

      11. expm1-define 97.3%

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

   3. Simplified 97.3%

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

   5. Taylor expanded in y around 0 77.2%

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

      1. associate-/l* 77.1%

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

      2. expm1-define 91.2%

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

   7. Simplified 91.2%

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

4. Final simplification 88.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.65 \cdot 10^{+62}:\\ \;\;\;\;x + \frac{-1}{\frac{0.5 \cdot \left(y \cdot t\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 82.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+65}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{0.5 \cdot \left(y \cdot t\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)}}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4e+65)
   x
   (+
    x
    (/
     -1.0
     (/
      (+
       (* 0.5 (* y t))
       (/ t (* z (+ 1.0 (* z (+ 0.5 (* z 0.16666666666666666)))))))
      y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4e+65) {
		tmp = x;
	} else {
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	}
	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 <= (-4d+65)) then
        tmp = x
    else
        tmp = x + ((-1.0d0) / (((0.5d0 * (y * t)) + (t / (z * (1.0d0 + (z * (0.5d0 + (z * 0.16666666666666666d0))))))) / y))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4e+65) {
		tmp = x;
	} else {
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4e+65:
		tmp = x
	else:
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4e+65)
		tmp = x;
	else
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(Float64(0.5 * Float64(y * t)) + Float64(t / Float64(z * Float64(1.0 + Float64(z * Float64(0.5 + Float64(z * 0.16666666666666666))))))) / y)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -4e+65)
		tmp = x;
	else
		tmp = x + (-1.0 / (((0.5 * (y * t)) + (t / (z * (1.0 + (z * (0.5 + (z * 0.16666666666666666))))))) / y));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4e+65], x, N[(x + N[(-1.0 / N[(N[(N[(0.5 * N[(y * t), $MachinePrecision]), $MachinePrecision] + N[(t / N[(z * N[(1.0 + N[(z * N[(0.5 + N[(z * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4e65

   1. Initial program 76.8%

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

      1. associate-+l- 76.8%

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

      2. sub-neg 76.8%

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

      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 58.2%

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

   if -4e65 < z

   1. Initial program 55.8%

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

      1. associate-+l- 73.1%

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

      2. sub-neg 73.1%

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

      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 97.0%

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

   3. Simplified 97.0%

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

      1. clear-num 97.0%

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

      2. associate-/r/ 97.0%

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

   6. Applied egg-rr 97.0%

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

      1. associate-*l/ 97.0%

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

      2. *-un-lft-identity 97.0%

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

      3. clear-num 97.0%

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

   8. Applied egg-rr 97.0%

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

   9. Taylor expanded in z around 0 92.4%

      \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(\color{blue}{z \cdot \left(y + z \cdot \left(0.16666666666666666 \cdot \left(y \cdot z\right) + 0.5 \cdot y\right)\right)}\right)}} \]

   10. Taylor expanded in y around 0 89.7%

       \[\leadsto x - \frac{1}{\color{blue}{\frac{0.5 \cdot \left(t \cdot y\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + 0.16666666666666666 \cdot z\right)\right)}}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{+65}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{0.5 \cdot \left(y \cdot t\right) + \frac{t}{z \cdot \left(1 + z \cdot \left(0.5 + z \cdot 0.16666666666666666\right)\right)}}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.6% accurate, 8.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-183}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-297} \lor \neg \left(t \leq 4.8 \cdot 10^{-280}\right) \land t \leq 4.1 \cdot 10^{-153}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -7.5e-183)
   x
   (if (or (<= t 3.5e-297) (and (not (<= t 4.8e-280)) (<= t 4.1e-153)))
     (* y (/ z (- t)))
     x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -7.5e-183) {
		tmp = x;
	} else if ((t <= 3.5e-297) || (!(t <= 4.8e-280) && (t <= 4.1e-153))) {
		tmp = y * (z / -t);
	} 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 <= (-7.5d-183)) then
        tmp = x
    else if ((t <= 3.5d-297) .or. (.not. (t <= 4.8d-280)) .and. (t <= 4.1d-153)) then
        tmp = y * (z / -t)
    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 <= -7.5e-183) {
		tmp = x;
	} else if ((t <= 3.5e-297) || (!(t <= 4.8e-280) && (t <= 4.1e-153))) {
		tmp = y * (z / -t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -7.5e-183:
		tmp = x
	elif (t <= 3.5e-297) or (not (t <= 4.8e-280) and (t <= 4.1e-153)):
		tmp = y * (z / -t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -7.5e-183)
		tmp = x;
	elseif ((t <= 3.5e-297) || (!(t <= 4.8e-280) && (t <= 4.1e-153)))
		tmp = Float64(y * Float64(z / Float64(-t)));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -7.5e-183)
		tmp = x;
	elseif ((t <= 3.5e-297) || (~((t <= 4.8e-280)) && (t <= 4.1e-153)))
		tmp = y * (z / -t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -7.5e-183], x, If[Or[LessEqual[t, 3.5e-297], And[N[Not[LessEqual[t, 4.8e-280]], $MachinePrecision], LessEqual[t, 4.1e-153]]], N[(y * N[(z / (-t)), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -7.5000000000000004e-183 or 3.4999999999999999e-297 < t < 4.7999999999999996e-280 or 4.1e-153 < t

   1. Initial program 69.3%

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

      1. associate-+l- 84.9%

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

      2. sub-neg 84.9%

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

      3. log1p-define 89.7%

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

      4. neg-sub0 89.7%

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

      5. associate-+l- 89.7%

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

      6. neg-sub0 89.7%

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

      7. +-commutative 89.7%

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

      8. unsub-neg 89.7%

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

      9. *-rgt-identity 89.7%

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

      10. distribute-lft-out-- 89.7%

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

      11. expm1-define 99.0%

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

   3. Simplified 99.0%

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

   5. Taylor expanded in x around inf 79.8%

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

   if -7.5000000000000004e-183 < t < 3.4999999999999999e-297 or 4.7999999999999996e-280 < t < 4.1e-153

   1. Initial program 28.8%

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

      1. associate-+l- 31.6%

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

      2. sub-neg 31.6%

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

      3. log1p-define 46.4%

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

      4. neg-sub0 46.4%

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

      5. associate-+l- 46.4%

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

      6. neg-sub0 46.4%

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

      7. +-commutative 46.4%

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

      8. unsub-neg 46.4%

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

      9. *-rgt-identity 46.4%

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

      10. distribute-lft-out-- 46.4%

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

      11. expm1-define 92.8%

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

   3. Simplified 92.8%

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

   5. Taylor expanded in x around 0 15.7%

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

      1. mul-1-neg 15.7%

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

      2. log1p-define 30.4%

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

      3. expm1-define 76.2%

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

      4. distribute-frac-neg2 76.2%

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

   7. Simplified 76.2%

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

   8. Taylor expanded in z around 0 44.0%

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

      1. mul-1-neg 44.0%

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

      2. associate-*r/ 50.8%

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

      3. distribute-lft-neg-in 50.8%

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

   10. Simplified 50.8%

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

4. Final simplification 73.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.5 \cdot 10^{-183}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-297} \lor \neg \left(t \leq 4.8 \cdot 10^{-280}\right) \land t \leq 4.1 \cdot 10^{-153}:\\ \;\;\;\;y \cdot \frac{z}{-t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 82.2% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.42:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z -0.42) x (- x (/ y (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.42) {
		tmp = x;
	} else {
		tmp = x - (y / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-0.42d0)) then
        tmp = x
    else
        tmp = x - (y / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.42) {
		tmp = x;
	} else {
		tmp = x - (y / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -0.42:
		tmp = x
	else:
		tmp = x - (y / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.42)
		tmp = x;
	else
		tmp = Float64(x - Float64(y / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -0.42)
		tmp = x;
	else
		tmp = x - (y / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.42], x, N[(x - N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.419999999999999984

   1. Initial program 78.7%

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

      1. associate-+l- 78.7%

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

      2. sub-neg 78.7%

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

      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 59.4%

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

   if -0.419999999999999984 < z

   1. Initial program 52.7%

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

      1. associate-+l- 71.9%

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

      2. sub-neg 71.9%

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

      3. log1p-define 71.9%

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

      4. neg-sub0 71.9%

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

      5. associate-+l- 71.9%

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

      6. neg-sub0 71.9%

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

      7. +-commutative 71.9%

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

      8. unsub-neg 71.9%

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

      9. *-rgt-identity 71.9%

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

      10. distribute-lft-out-- 71.9%

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

      11. expm1-define 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in z around 0 89.2%

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

      1. associate-/l* 91.8%

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

   7. Simplified 91.8%

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

      1. clear-num 91.8%

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

      2. un-div-inv 91.8%

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

   9. Applied egg-rr 91.8%

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

Alternative 11: 82.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z -3.6) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.6) {
		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 <= (-3.6d0)) 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 <= -3.6) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.6:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.6)
		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 <= -3.6)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.6], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.60000000000000009

   1. Initial program 78.7%

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

      1. associate-+l- 78.7%

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

      2. sub-neg 78.7%

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

      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 59.4%

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

   if -3.60000000000000009 < z

   1. Initial program 52.7%

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

      1. associate-+l- 71.9%

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

      2. sub-neg 71.9%

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

      3. log1p-define 71.9%

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

      4. neg-sub0 71.9%

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

      5. associate-+l- 71.9%

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

      6. neg-sub0 71.9%

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

      7. +-commutative 71.9%

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

      8. unsub-neg 71.9%

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

      9. *-rgt-identity 71.9%

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

      10. distribute-lft-out-- 71.9%

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

      11. expm1-define 96.7%

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

   3. Simplified 96.7%

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

   5. Taylor expanded in z around 0 89.2%

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

      1. associate-/l* 91.8%

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

   7. Simplified 91.8%

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

Alternative 12: 71.5% 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 61.1%

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

   1. associate-+l- 74.1%

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

   2. sub-neg 74.1%

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

   3. log1p-define 80.9%

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

   4. neg-sub0 80.9%

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

   5. associate-+l- 80.9%

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

   6. neg-sub0 80.9%

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

   7. +-commutative 80.9%

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

   8. unsub-neg 80.9%

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

   9. *-rgt-identity 80.9%

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

   10. distribute-lft-out-- 80.9%

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

   11. expm1-define 97.7%

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

3. Simplified 97.7%

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

5. Taylor expanded in x around inf 67.4%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Developer target: 74.5% 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 2024096 
(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)))