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

Percentage Accurate: 62.6% → 98.2%

Time: 12.8s

Alternatives: 5

Speedup: 211.0×

• could not determine a ground truth

  1. x = 4.794488321091632e+96
  2. y = 2.8515826645142764e+285
  3. z = 9.822525291391147e+234
  4. t = 6.84382978694979e-96

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: 62.6% 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: 98.2% 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 60.3%

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

   1. associate-+l- 75.9%

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

   2. sub-neg 75.9%

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

   3. log1p-def 81.8%

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

   4. neg-sub0 81.8%

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

   5. associate-+l- 81.8%

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

   6. neg-sub0 81.8%

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

   7. neg-mul-1 81.8%

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

   8. *-commutative 81.8%

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

   9. distribute-rgt-out 81.8%

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

   10. +-commutative 81.8%

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

   11. metadata-eval 81.8%

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

   12. sub-neg 81.8%

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

   13. expm1-def 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. Final simplification 97.3%

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

Alternative 2: 86.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.3e+24) x (- x (* y (/ (expm1 z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.3e+24) {
		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 (x <= -1.3e+24) {
		tmp = x;
	} else {
		tmp = x - (y * (Math.expm1(z) / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -1.3e+24:
		tmp = x
	else:
		tmp = x - (y * (math.expm1(z) / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.3e+24)
		tmp = x;
	else
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.3e+24], x, N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2999999999999999e24

   1. Initial program 79.1%

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

      1. associate-+l- 95.7%

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

      2. sub-neg 95.7%

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

      3. log1p-def 97.9%

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

      4. neg-sub0 97.9%

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

      5. associate-+l- 97.9%

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

      6. neg-sub0 97.9%

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

      7. neg-mul-1 97.9%

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

      8. *-commutative 97.9%

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

      9. distribute-rgt-out 97.9%

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

      10. +-commutative 97.9%

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

      11. metadata-eval 97.9%

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

      12. sub-neg 97.9%

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

      13. expm1-def 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. Taylor expanded in z around 0 72.9%

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

      1. associate-/l* 72.9%

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

      2. associate-/r/ 71.2%

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

   6. Simplified 71.2%

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

      1. associate-*l/ 72.9%

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

      2. associate-/l* 72.9%

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

   8. Applied egg-rr 72.9%

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

   9. Taylor expanded in x around inf 89.9%

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

   if -1.2999999999999999e24 < x

   1. Initial program 54.9%

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

      1. associate-+l- 70.3%

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

      2. sub-neg 70.3%

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

      3. log1p-def 77.2%

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

      4. neg-sub0 77.2%

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

      5. associate-+l- 77.2%

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

      6. neg-sub0 77.2%

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

      7. neg-mul-1 77.2%

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

      8. *-commutative 77.2%

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

      9. distribute-rgt-out 77.2%

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

      10. +-commutative 77.2%

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

      11. metadata-eval 77.2%

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

      12. sub-neg 77.2%

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

      13. expm1-def 96.5%

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

   3. Simplified 96.5%

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

   4. Taylor expanded in y around 0 72.1%

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

      1. associate-/l* 71.6%

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

      2. associate-/r/ 72.1%

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

      3. expm1-def 90.5%

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

   6. Simplified 90.5%

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

4. Final simplification 90.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.3 \cdot 10^{+24}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \]

Alternative 3: 80.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.6e-11) x (- x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.6e-11) {
		tmp = x;
	} else {
		tmp = x - (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) :: tmp
    if (z <= (-3.6d-11)) then
        tmp = x
    else
        tmp = x - (z * (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.6e-11) {
		tmp = x;
	} else {
		tmp = x - (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.6e-11:
		tmp = x
	else:
		tmp = x - (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.6e-11)
		tmp = x;
	else
		tmp = Float64(x - Float64(z * Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.6e-11)
		tmp = x;
	else
		tmp = x - (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.6e-11], x, N[(x - N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.59999999999999985e-11

   1. Initial program 81.3%

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

      1. associate-+l- 81.3%

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

      2. sub-neg 81.3%

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

      3. log1p-def 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate-+l- 99.8%

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

      6. neg-sub0 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-rgt-out 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. sub-neg 99.8%

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

      13. expm1-def 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. Taylor expanded in z around 0 43.4%

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

      1. associate-/l* 40.8%

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

      2. associate-/r/ 43.5%

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

   6. Simplified 43.5%

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

      1. associate-*l/ 43.4%

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

      2. associate-/l* 40.8%

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

   8. Applied egg-rr 40.8%

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

   9. Taylor expanded in x around inf 65.9%

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

   if -3.59999999999999985e-11 < z

   1. Initial program 50.7%

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

      1. associate-+l- 73.5%

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

      2. sub-neg 73.5%

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

      3. log1p-def 73.6%

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

      4. neg-sub0 73.6%

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

      5. associate-+l- 73.6%

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

      6. neg-sub0 73.6%

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

      7. neg-mul-1 73.6%

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

      8. *-commutative 73.6%

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

      9. distribute-rgt-out 73.7%

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

      10. +-commutative 73.7%

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

      11. metadata-eval 73.7%

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

      12. sub-neg 73.7%

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

      13. expm1-def 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in z around 0 90.0%

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

      1. associate-/l* 94.1%

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

      2. associate-/r/ 89.9%

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

   6. Simplified 89.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 4: 82.8% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.4e-11) x (- x (/ y (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.4e-11) {
		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 <= (-4.4d-11)) 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 <= -4.4e-11) {
		tmp = x;
	} else {
		tmp = x - (y / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4.4e-11:
		tmp = x
	else:
		tmp = x - (y / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.4e-11)
		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 <= -4.4e-11)
		tmp = x;
	else
		tmp = x - (y / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.4e-11], x, N[(x - N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.4000000000000003e-11

   1. Initial program 81.3%

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

      1. associate-+l- 81.3%

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

      2. sub-neg 81.3%

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

      3. log1p-def 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate-+l- 99.8%

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

      6. neg-sub0 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-rgt-out 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. sub-neg 99.8%

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

      13. expm1-def 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. Taylor expanded in z around 0 43.4%

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

      1. associate-/l* 40.8%

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

      2. associate-/r/ 43.5%

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

   6. Simplified 43.5%

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

      1. associate-*l/ 43.4%

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

      2. associate-/l* 40.8%

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

   8. Applied egg-rr 40.8%

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

   9. Taylor expanded in x around inf 65.9%

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

   if -4.4000000000000003e-11 < z

   1. Initial program 50.7%

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

      1. associate-+l- 73.5%

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

      2. sub-neg 73.5%

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

      3. log1p-def 73.6%

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

      4. neg-sub0 73.6%

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

      5. associate-+l- 73.6%

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

      6. neg-sub0 73.6%

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

      7. neg-mul-1 73.6%

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

      8. *-commutative 73.6%

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

      9. distribute-rgt-out 73.7%

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

      10. +-commutative 73.7%

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

      11. metadata-eval 73.7%

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

      12. sub-neg 73.7%

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

      13. expm1-def 96.1%

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

   3. Simplified 96.1%

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

   4. Taylor expanded in z around 0 90.0%

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

      1. associate-/l* 94.1%

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

      2. associate-/r/ 89.9%

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

   6. Simplified 89.9%

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

      1. associate-*l/ 90.0%

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

      2. associate-/l* 94.1%

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

   8. Applied egg-rr 94.1%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.4 \cdot 10^{-11}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 5: 72.3% 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 60.3%

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

   1. associate-+l- 75.9%

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

   2. sub-neg 75.9%

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

   3. log1p-def 81.8%

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

   4. neg-sub0 81.8%

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

   5. associate-+l- 81.8%

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

   6. neg-sub0 81.8%

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

   7. neg-mul-1 81.8%

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

   8. *-commutative 81.8%

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

   9. distribute-rgt-out 81.8%

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

   10. +-commutative 81.8%

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

   11. metadata-eval 81.8%

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

   12. sub-neg 81.8%

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

   13. expm1-def 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. Taylor expanded in z around 0 75.5%

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

   1. associate-/l* 77.4%

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

   2. associate-/r/ 75.4%

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

6. Simplified 75.4%

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

   1. associate-*l/ 75.5%

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

   2. associate-/l* 77.4%

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

8. Applied egg-rr 77.4%

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

9. Taylor expanded in x around inf 70.6%

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

10. Final simplification 70.6%

    \[\leadsto x \]

Developer target: 75.1% 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 2023173 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :herbie-target
  (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)))