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

Percentage Accurate: 60.3% → 98.6%

Time: 17.5s

Alternatives: 8

Speedup: 211.0×

• Could not converge on a ground truth

  1. x = 2.0778075021794823e+79
  2. y = 1.2278821451952227e-229
  3. z = 4.868109632274046e+93
  4. t = 1.862110283891484e-275

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 60.3% 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: 98.6% 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 59.4%

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

   1. associate-+l- 73.2%

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

   2. sub-neg 73.2%

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

   3. log1p-define 84.1%

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

   4. neg-sub0 84.1%

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

   5. associate-+l- 84.1%

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

   6. neg-sub0 84.1%

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

   7. +-commutative 84.1%

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

   8. unsub-neg 84.1%

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

   9. *-rgt-identity 84.1%

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

   10. distribute-lft-out-- 84.1%

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

   11. expm1-define 98.8%

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

3. Simplified 98.8%

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

Alternative 2: 91.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.02:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + z \cdot \left(y \cdot 0.5 + z \cdot \left(0.041666666666666664 \cdot \left(y \cdot z\right) + y \cdot 0.16666666666666666\right)\right)\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.02)
   (- x (/ (* y (expm1 z)) t))
   (-
    x
    (/
     (log1p
      (*
       z
       (+
        y
        (*
         z
         (+
          (* y 0.5)
          (*
           z
           (+ (* 0.041666666666666664 (* y z)) (* y 0.16666666666666666))))))))
     t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.02) {
		tmp = x - ((y * expm1(z)) / t);
	} else {
		tmp = x - (log1p((z * (y + (z * ((y * 0.5) + (z * ((0.041666666666666664 * (y * z)) + (y * 0.16666666666666666)))))))) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.02) {
		tmp = x - ((y * Math.expm1(z)) / t);
	} else {
		tmp = x - (Math.log1p((z * (y + (z * ((y * 0.5) + (z * ((0.041666666666666664 * (y * z)) + (y * 0.16666666666666666)))))))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -0.02:
		tmp = x - ((y * math.expm1(z)) / t)
	else:
		tmp = x - (math.log1p((z * (y + (z * ((y * 0.5) + (z * ((0.041666666666666664 * (y * z)) + (y * 0.16666666666666666)))))))) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.02)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	else
		tmp = Float64(x - Float64(log1p(Float64(z * Float64(y + Float64(z * Float64(Float64(y * 0.5) + Float64(z * Float64(Float64(0.041666666666666664 * Float64(y * z)) + Float64(y * 0.16666666666666666)))))))) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.02], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(z * N[(y + N[(z * N[(N[(y * 0.5), $MachinePrecision] + N[(z * N[(N[(0.041666666666666664 * N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(y * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0200000000000000004

   1. Initial program 66.7%

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

      1. associate-+l- 66.7%

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

      2. sub-neg 66.7%

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

      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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 89.7%

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

      1. expm1-define 89.7%

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

   7. Simplified 89.7%

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

   if -0.0200000000000000004 < z

   1. Initial program 56.1%

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

      1. associate-+l- 76.2%

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

      2. sub-neg 76.2%

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

      3. log1p-define 77.1%

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

      4. neg-sub0 77.1%

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

      5. associate-+l- 77.1%

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

      6. neg-sub0 77.1%

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

      7. +-commutative 77.1%

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

      8. unsub-neg 77.1%

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

      9. *-rgt-identity 77.1%

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

      10. distribute-lft-out-- 77.1%

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

      11. expm1-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in z around 0 98.4%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{z \cdot \left(y + z \cdot \left(0.5 \cdot y + z \cdot \left(0.041666666666666664 \cdot \left(y \cdot z\right) + 0.16666666666666666 \cdot y\right)\right)\right)}\right)}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 95.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.02:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + z \cdot \left(y \cdot 0.5 + z \cdot \left(0.041666666666666664 \cdot \left(y \cdot z\right) + y \cdot 0.16666666666666666\right)\right)\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.5% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.0082:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + z \cdot \left(y \cdot 0.5 + \left(y \cdot z\right) \cdot 0.16666666666666666\right)\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.0082)
   (- x (/ (* y (expm1 z)) t))
   (-
    x
    (/
     (log1p (* z (+ y (* z (+ (* y 0.5) (* (* y z) 0.16666666666666666))))))
     t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.0082) {
		tmp = x - ((y * expm1(z)) / t);
	} else {
		tmp = x - (log1p((z * (y + (z * ((y * 0.5) + ((y * z) * 0.16666666666666666)))))) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.0082) {
		tmp = x - ((y * Math.expm1(z)) / t);
	} else {
		tmp = x - (Math.log1p((z * (y + (z * ((y * 0.5) + ((y * z) * 0.16666666666666666)))))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -0.0082:
		tmp = x - ((y * math.expm1(z)) / t)
	else:
		tmp = x - (math.log1p((z * (y + (z * ((y * 0.5) + ((y * z) * 0.16666666666666666)))))) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.0082)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	else
		tmp = Float64(x - Float64(log1p(Float64(z * Float64(y + Float64(z * Float64(Float64(y * 0.5) + Float64(Float64(y * z) * 0.16666666666666666)))))) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.0082], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(z * N[(y + N[(z * N[(N[(y * 0.5), $MachinePrecision] + N[(N[(y * z), $MachinePrecision] * 0.16666666666666666), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.00820000000000000069

   1. Initial program 66.7%

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

      1. associate-+l- 66.7%

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

      2. sub-neg 66.7%

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

      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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 89.7%

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

      1. expm1-define 89.7%

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

   7. Simplified 89.7%

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

   if -0.00820000000000000069 < z

   1. Initial program 56.1%

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

      1. associate-+l- 76.2%

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

      2. sub-neg 76.2%

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

      3. log1p-define 77.1%

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

      4. neg-sub0 77.1%

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

      5. associate-+l- 77.1%

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

      6. neg-sub0 77.1%

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

      7. +-commutative 77.1%

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

      8. unsub-neg 77.1%

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

      9. *-rgt-identity 77.1%

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

      10. distribute-lft-out-- 77.1%

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

      11. expm1-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in z around 0 98.3%

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

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -0.0082:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + z \cdot \left(y \cdot 0.5 + \left(y \cdot z\right) \cdot 0.16666666666666666\right)\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 91.4% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.00195:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(z \cdot \left(y + 0.5 \cdot \left(y \cdot z\right)\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.00195)
   (- x (/ (* y (expm1 z)) t))
   (- x (/ (log1p (* z (+ y (* 0.5 (* y z))))) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.00195) {
		tmp = x - ((y * expm1(z)) / t);
	} 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 <= -0.00195) {
		tmp = x - ((y * Math.expm1(z)) / t);
	} 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 <= -0.00195:
		tmp = x - ((y * math.expm1(z)) / t)
	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 <= -0.00195)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	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, -0.00195], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $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 < -0.0019499999999999999

   1. Initial program 66.7%

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

      1. associate-+l- 66.7%

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

      2. sub-neg 66.7%

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

      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 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 89.7%

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

      1. expm1-define 89.7%

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

   7. Simplified 89.7%

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

   if -0.0019499999999999999 < z

   1. Initial program 56.1%

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

      1. associate-+l- 76.2%

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

      2. sub-neg 76.2%

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

      3. log1p-define 77.1%

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

      4. neg-sub0 77.1%

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

      5. associate-+l- 77.1%

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

      6. neg-sub0 77.1%

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

      7. +-commutative 77.1%

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

      8. unsub-neg 77.1%

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

      9. *-rgt-identity 77.1%

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

      10. distribute-lft-out-- 77.1%

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

      11. expm1-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in z around 0 97.9%

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

Alternative 5: 85.8% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.4%

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

   1. associate-+l- 73.2%

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

   2. sub-neg 73.2%

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

   3. log1p-define 84.1%

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

   4. neg-sub0 84.1%

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

   5. associate-+l- 84.1%

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

   6. neg-sub0 84.1%

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

   7. +-commutative 84.1%

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

   8. unsub-neg 84.1%

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

   9. *-rgt-identity 84.1%

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

   10. distribute-lft-out-- 84.1%

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

   11. expm1-define 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in y around 0 80.3%

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

   1. associate-/l* 80.2%

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

   2. expm1-define 90.5%

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

7. Simplified 90.5%

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

Alternative 6: 81.7% accurate, 9.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{-1}{\frac{-0.5 \cdot \frac{z \cdot t}{y} + \frac{t}{y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.8e-15)
   (+ x (/ -1.0 (/ (+ (* -0.5 (/ (* z t) y)) (/ t y)) z)))
   (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.8e-15) {
		tmp = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z));
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-5.8d-15)) then
        tmp = x + ((-1.0d0) / ((((-0.5d0) * ((z * t) / y)) + (t / y)) / z))
    else
        tmp = x - (y * (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.8e-15) {
		tmp = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z));
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5.8e-15:
		tmp = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z))
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.8e-15)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(Float64(-0.5 * Float64(Float64(z * t) / y)) + Float64(t / y)) / z)));
	else
		tmp = Float64(x - Float64(y * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -5.8e-15)
		tmp = x + (-1.0 / (((-0.5 * ((z * t) / y)) + (t / y)) / z));
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.8e-15], N[(x + N[(-1.0 / N[(N[(N[(-0.5 * N[(N[(z * t), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.80000000000000037e-15

   1. Initial program 66.7%

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

      1. associate-+l- 66.7%

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

      2. sub-neg 66.7%

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

      3. log1p-define 98.6%

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

      4. neg-sub0 98.6%

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

      5. associate-+l- 98.6%

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

      6. neg-sub0 98.6%

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

      7. +-commutative 98.6%

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

      8. unsub-neg 98.6%

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

      9. *-rgt-identity 98.6%

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

      10. distribute-lft-out-- 98.6%

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

      11. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in y around 0 88.0%

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

      1. expm1-define 89.3%

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

   7. Simplified 89.3%

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

      1. clear-num 89.1%

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

      2. inv-pow 89.1%

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

   9. Applied egg-rr 89.1%

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

       1. unpow-1 89.1%

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

       2. associate-/r* 89.1%

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

   11. Simplified 89.1%

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

   12. Taylor expanded in z around 0 63.6%

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

   if -5.80000000000000037e-15 < z

   1. Initial program 55.7%

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

      1. associate-+l- 76.5%

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

      2. sub-neg 76.5%

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

      3. log1p-define 76.9%

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

      4. neg-sub0 76.9%

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

      5. associate-+l- 76.9%

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

      6. neg-sub0 76.9%

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

      7. +-commutative 76.9%

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

      8. unsub-neg 76.9%

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

      9. *-rgt-identity 76.9%

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

      10. distribute-lft-out-- 76.9%

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

      11. expm1-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in z around 0 89.2%

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

      1. mul-1-neg 89.2%

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

      2. unsub-neg 89.2%

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

      3. associate-/l* 90.8%

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

   7. Simplified 90.8%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{-1}{\frac{-0.5 \cdot \frac{z \cdot t}{y} + \frac{t}{y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.9% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -450:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z -450.0) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -450.0) {
		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 <= (-450.0d0)) 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 <= -450.0) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -450.0:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -450.0)
		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 <= -450.0)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -450.0], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -450

   1. Initial program 67.8%

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

      1. associate-+l- 67.8%

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

      2. sub-neg 67.8%

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

      3. log1p-define 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. neg-sub0 100.0%

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

      7. +-commutative 100.0%

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

      8. unsub-neg 100.0%

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

      9. *-rgt-identity 100.0%

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

      10. distribute-lft-out-- 100.0%

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

      11. expm1-define 100.0%

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

   3. Simplified 100.0%

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

   5. Taylor expanded in x around inf 59.9%

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

   if -450 < 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- 75.6%

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

      2. sub-neg 75.6%

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

      3. log1p-define 77.4%

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

      4. neg-sub0 77.4%

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

      5. associate-+l- 77.4%

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

      6. neg-sub0 77.4%

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

      7. +-commutative 77.4%

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

      8. unsub-neg 77.4%

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

      9. *-rgt-identity 77.4%

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

      10. distribute-lft-out-- 77.4%

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

      11. expm1-define 98.3%

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

   3. Simplified 98.3%

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

   5. Taylor expanded in z around 0 88.0%

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

      1. mul-1-neg 88.0%

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

      2. unsub-neg 88.0%

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

      3. associate-/l* 89.5%

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

   7. Simplified 89.5%

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

Alternative 8: 70.4% 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 59.4%

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

   1. associate-+l- 73.2%

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

   2. sub-neg 73.2%

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

   3. log1p-define 84.1%

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

   4. neg-sub0 84.1%

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

   5. associate-+l- 84.1%

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

   6. neg-sub0 84.1%

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

   7. +-commutative 84.1%

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

   8. unsub-neg 84.1%

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

   9. *-rgt-identity 84.1%

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

   10. distribute-lft-out-- 84.1%

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

   11. expm1-define 98.8%

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

3. Simplified 98.8%

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

5. Taylor expanded in x around inf 69.8%

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

Developer Target 1: 74.1% 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 2024170 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -288746230882079470000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (- x (/ (/ (- 1/2) (* y t)) (* z z))) (* (/ (- 1/2) (* y t)) (/ (/ 2 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t))))

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