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

Percentage Accurate: 60.8% → 98.5%

Time: 21.9s

Alternatives: 6

Speedup: 211.0×

• could not determine a ground truth

  1. x = 3.005598328074984e+44
  2. y = 4.473235407907223e-233
  3. z = 6.895471859752721e+242
  4. t = 3.0566307480694176e-65

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: 60.8% 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: 98.5% 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 64.3%

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

   1. sub-neg 64.3%

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

   2. associate-+l+ 78.0%

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

   3. cancel-sign-sub 78.0%

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

   4. log1p-define 82.0%

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

   5. cancel-sign-sub 82.0%

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

   6. +-commutative 82.0%

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

   7. unsub-neg 82.0%

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

   8. *-rgt-identity 82.0%

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

   9. distribute-lft-out-- 82.0%

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

   10. 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. Final simplification 97.7%

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

Alternative 2: 89.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ x + \frac{\frac{-1}{t}}{0.5 + \frac{1}{y \cdot \mathsf{expm1}\left(z\right)}} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ x (/ (/ -1.0 t) (+ 0.5 (/ 1.0 (* y (expm1 z)))))))

C

double code(double x, double y, double z, double t) {
	return x + ((-1.0 / t) / (0.5 + (1.0 / (y * expm1(z)))));
}

Java

public static double code(double x, double y, double z, double t) {
	return x + ((-1.0 / t) / (0.5 + (1.0 / (y * Math.expm1(z)))));
}

Python

def code(x, y, z, t):
	return x + ((-1.0 / t) / (0.5 + (1.0 / (y * math.expm1(z)))))

Julia

function code(x, y, z, t)
	return Float64(x + Float64(Float64(-1.0 / t) / Float64(0.5 + Float64(1.0 / Float64(y * expm1(z))))))
end

Wolfram

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

Derivation

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. sub-neg 64.3%

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

   2. associate-+l+ 78.0%

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

   3. cancel-sign-sub 78.0%

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

   4. log1p-define 82.0%

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

   5. cancel-sign-sub 82.0%

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

   6. +-commutative 82.0%

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

   7. unsub-neg 82.0%

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

   8. *-rgt-identity 82.0%

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

   9. distribute-lft-out-- 82.0%

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

   10. 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. Step-by-step derivation

   1. clear-num 97.7%

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

   2. inv-pow 97.7%

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

6. Applied egg-rr 97.7%

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

   1. clear-num 97.7%

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

   2. associate-/r/ 97.6%

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

8. Applied egg-rr 97.6%

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

   1. unpow-1 97.6%

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

   2. *-commutative 97.6%

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

   3. associate-/r* 97.6%

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

10. Applied egg-rr 97.6%

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

11. Taylor expanded in y around 0 78.2%

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

    1. expm1-define 88.8%

       \[\leadsto x - \frac{\frac{1}{t}}{0.5 + \frac{1}{y \cdot \color{blue}{\mathsf{expm1}\left(z\right)}}} \]

13. Simplified 88.8%

    \[\leadsto x - \frac{\frac{1}{t}}{\color{blue}{0.5 + \frac{1}{y \cdot \mathsf{expm1}\left(z\right)}}} \]

14. Final simplification 88.8%

    \[\leadsto x + \frac{\frac{-1}{t}}{0.5 + \frac{1}{y \cdot \mathsf{expm1}\left(z\right)}} \]
15. Add Preprocessing

Alternative 3: 86.5% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+142}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.2e+142) x (- x (* y (/ (expm1 z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.2e+142) {
		tmp = x;
	} 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 <= -7.2e+142) {
		tmp = x;
	} else {
		tmp = x - (y * (Math.expm1(z) / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -7.2e+142:
		tmp = x
	else:
		tmp = x - (y * (math.expm1(z) / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.2e+142)
		tmp = x;
	else
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.2e+142], x, N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -7.2000000000000003e142

   1. Initial program 63.8%

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

      1. sub-neg 63.8%

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

      2. associate-+l+ 81.3%

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

      3. cancel-sign-sub 81.3%

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

      4. log1p-define 81.3%

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

      5. cancel-sign-sub 81.3%

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

      6. +-commutative 81.3%

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

      7. unsub-neg 81.3%

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

      8. *-rgt-identity 81.3%

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

      9. distribute-lft-out-- 81.6%

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

      10. 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. Taylor expanded in x around inf 59.0%

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

   if -7.2000000000000003e142 < y

   1. Initial program 64.4%

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

      1. sub-neg 64.4%

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

      2. associate-+l+ 77.5%

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

      3. cancel-sign-sub 77.5%

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

      4. log1p-define 82.1%

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

      5. cancel-sign-sub 82.1%

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

      6. +-commutative 82.1%

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

      7. unsub-neg 82.1%

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

      8. *-rgt-identity 82.1%

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

      9. distribute-lft-out-- 82.1%

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

      10. expm1-define 97.5%

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

   3. Simplified 97.5%

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

   5. Taylor expanded in y around 0 80.0%

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

      1. expm1-define 90.4%

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

      2. associate-/l* 91.1%

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

   7. Simplified 91.1%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -7.2 \cdot 10^{+142}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 81.3% accurate, 15.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \frac{-1}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6.2e-31) x (+ x (* y (* z (/ -1.0 t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.2e-31) {
		tmp = x;
	} else {
		tmp = x + (y * (z * (-1.0 / t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-6.2d-31)) then
        tmp = x
    else
        tmp = x + (y * (z * ((-1.0d0) / t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.2e-31) {
		tmp = x;
	} else {
		tmp = x + (y * (z * (-1.0 / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -6.2e-31:
		tmp = x
	else:
		tmp = x + (y * (z * (-1.0 / t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6.2e-31)
		tmp = x;
	else
		tmp = Float64(x + Float64(y * Float64(z * Float64(-1.0 / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -6.2e-31)
		tmp = x;
	else
		tmp = x + (y * (z * (-1.0 / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6.2e-31], x, N[(x + N[(y * N[(z * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.19999999999999999e-31

   1. Initial program 85.4%

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

      1. sub-neg 85.4%

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

      2. associate-+l+ 85.6%

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

      3. cancel-sign-sub 85.6%

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

      4. log1p-define 97.2%

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

      5. cancel-sign-sub 97.2%

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

      6. +-commutative 97.2%

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

      7. unsub-neg 97.2%

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

      8. *-rgt-identity 97.2%

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

      9. distribute-lft-out-- 97.4%

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

      10. 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 72.3%

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

   if -6.19999999999999999e-31 < z

   1. Initial program 54.2%

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

      1. sub-neg 54.2%

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

      2. associate-+l+ 74.3%

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

      3. cancel-sign-sub 74.3%

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

      4. log1p-define 74.7%

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

      5. cancel-sign-sub 74.7%

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

      6. +-commutative 74.7%

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

      7. unsub-neg 74.7%

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

      8. *-rgt-identity 74.7%

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

      9. distribute-lft-out-- 74.7%

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

      10. 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 87.8%

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

      1. frac-2neg 87.8%

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

      2. div-inv 87.7%

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

      3. distribute-rgt-neg-in 87.7%

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

      4. neg-mul-1 87.7%

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

      5. associate-/r* 87.7%

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

      6. metadata-eval 87.7%

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

   7. Applied egg-rr 87.7%

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

      1. associate-*l* 88.7%

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

   9. Simplified 88.7%

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

       1. frac-2neg 88.7%

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

       2. metadata-eval 88.7%

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

       3. div-inv 88.7%

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

       4. frac-2neg 88.7%

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

       5. clear-num 88.3%

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

   11. Applied egg-rr 88.3%

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

       1. associate-/r/ 88.7%

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

   13. Simplified 88.7%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \frac{-1}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.3% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.3e-31) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.3e-31) {
		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 <= (-1.3d-31)) 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 <= -1.3e-31) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.3e-31:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.3e-31)
		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 <= -1.3e-31)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.3e-31], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.29999999999999998e-31

   1. Initial program 85.4%

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

      1. sub-neg 85.4%

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

      2. associate-+l+ 85.6%

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

      3. cancel-sign-sub 85.6%

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

      4. log1p-define 97.2%

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

      5. cancel-sign-sub 97.2%

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

      6. +-commutative 97.2%

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

      7. unsub-neg 97.2%

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

      8. *-rgt-identity 97.2%

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

      9. distribute-lft-out-- 97.4%

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

      10. 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 72.3%

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

   if -1.29999999999999998e-31 < z

   1. Initial program 54.2%

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

      1. sub-neg 54.2%

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

      2. associate-+l+ 74.3%

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

      3. cancel-sign-sub 74.3%

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

      4. log1p-define 74.7%

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

      5. cancel-sign-sub 74.7%

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

      6. +-commutative 74.7%

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

      7. unsub-neg 74.7%

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

      8. *-rgt-identity 74.7%

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

      9. distribute-lft-out-- 74.7%

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

      10. 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 87.8%

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

      1. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

4. Final simplification 83.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.3 \cdot 10^{-31}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 70.8% 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 64.3%

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

   1. sub-neg 64.3%

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

   2. associate-+l+ 78.0%

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

   3. cancel-sign-sub 78.0%

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

   4. log1p-define 82.0%

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

   5. cancel-sign-sub 82.0%

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

   6. +-commutative 82.0%

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

   7. unsub-neg 82.0%

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

   8. *-rgt-identity 82.0%

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

   9. distribute-lft-out-- 82.0%

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

   10. 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 73.6%

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

6. Final simplification 73.6%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 74.3% 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 2024046 
(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)))