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

Percentage Accurate: 60.9% → 98.6%

Time: 14.8s

Alternatives: 8

Speedup: 11.3×

• could not determine a ground truth

  1. x = -1.1003373596567313e-250
  2. y = 4.002480287233177e+272
  3. z = 8.563064146887863e+186
  4. t = 3.562818475312156e+44

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: 60.9% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in z around inf

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

   1. associate--l+ N/A

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

   2. *-rgt-identity N/A

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

   3. distribute-lft-out-- N/A

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

   4. lower-log1p.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-expm1.f64 98.1

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

5. Applied rewrites 98.1%

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

Alternative 2: 88.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+148} \lor \neg \left(y \leq 1.36 \cdot 10^{+139}\right):\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -8.8e+148) (not (<= y 1.36e+139)))
   (- x (/ (log (fma z y 1.0)) t))
   (- x (* (/ (expm1 z) t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -8.8e+148) || !(y <= 1.36e+139)) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else {
		tmp = x - ((expm1(z) / t) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -8.8e+148) || !(y <= 1.36e+139))
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -8.8e+148], N[Not[LessEqual[y, 1.36e+139]], $MachinePrecision]], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.7999999999999995e148 or 1.36000000000000004e139 < y

   1. Initial program 22.1%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 74.4

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

   5. Applied rewrites 74.4%

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

   if -8.7999999999999995e148 < y < 1.36000000000000004e139

   1. Initial program 61.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 94.6

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

   5. Applied rewrites 94.6%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.8 \cdot 10^{+148} \lor \neg \left(y \leq 1.36 \cdot 10^{+139}\right):\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 81.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+50}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.8e+50) (- x (/ (log 1.0) t)) (fma (- y) (/ z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.8e+50) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = fma(-y, (z / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.8e+50)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = fma(Float64(-y), Float64(z / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.8e+50], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[((-y) * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.79999999999999987e50

   1. Initial program 72.4%

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

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 55.8%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]

   if -3.79999999999999987e50 < z

   1. Initial program 49.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. *-rgt-identity N/A

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

      3. distribute-lft-out-- N/A

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

      4. lower-log1p.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-expm1.f64 97.5

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

   5. Applied rewrites 97.5%

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

   6. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. distribute-lft-neg-in N/A

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

      5. lower-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(y\right), \frac{z}{t}, x\right)} \]

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 88.5

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

   8. Applied rewrites 88.5%

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

Alternative 4: 86.2% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. div-sub N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. div-sub N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 87.4

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

5. Applied rewrites 87.4%

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

Alternative 5: 74.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in z around inf

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

   1. associate--l+ N/A

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

   2. *-rgt-identity N/A

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

   3. distribute-lft-out-- N/A

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

   4. lower-log1p.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-expm1.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(y\right), \frac{z}{t}, x\right)} \]

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 74.6

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

8. Applied rewrites 74.6%

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

Alternative 6: 72.3% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

      \[\leadsto \color{blue}{z \cdot \left(\frac{-1}{2} \cdot \frac{z \cdot \left(y + -1 \cdot {y}^{2}\right)}{t} - \frac{y}{t}\right) + x} \]

   2. *-commutative N/A

      \[\leadsto \color{blue}{\left(\frac{-1}{2} \cdot \frac{z \cdot \left(y + -1 \cdot {y}^{2}\right)}{t} - \frac{y}{t}\right) \cdot z} + x \]

   3. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{-1}{2} \cdot \frac{z \cdot \left(y + -1 \cdot {y}^{2}\right)}{t} - \frac{y}{t}, z, x\right)} \]

5. Applied rewrites 62.9%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 73.4%

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

Alternative 7: 15.5% accurate, 11.9× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return -y * (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 = -y * (z / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in z around inf

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

   1. associate--l+ N/A

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

   2. *-rgt-identity N/A

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

   3. distribute-lft-out-- N/A

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

   4. lower-log1p.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-expm1.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(y\right), \frac{z}{t}, x\right)} \]

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 74.6

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

8. Applied rewrites 74.6%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 18.5%

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

Alternative 8: 13.5% accurate, 11.9× speedup

Math

\[\begin{array}{l} \\ \frac{-y}{t} \cdot z \end{array} \]

FPCore

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

C

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

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 = (-y / t) * z
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 54.5%

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

3. Taylor expanded in z around inf

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

   1. associate--l+ N/A

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

   2. *-rgt-identity N/A

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

   3. distribute-lft-out-- N/A

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

   4. lower-log1p.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower-expm1.f64 98.1

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

5. Applied rewrites 98.1%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. associate-/l* N/A

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

   4. distribute-lft-neg-in N/A

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

   5. lower-fma.f64 N/A

      \[\leadsto \color{blue}{\mathsf{fma}\left(\mathsf{neg}\left(y\right), \frac{z}{t}, x\right)} \]

   6. lower-neg.f64 N/A

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

   7. lower-/.f64 74.6

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

8. Applied rewrites 74.6%

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

9. Taylor expanded in x around 0

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

11. Applied rewrites 18.5%

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

13. Applied rewrites 17.5%

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

14. Final simplification 17.5%

    \[\leadsto \frac{-y}{t} \cdot z \]
15. Add Preprocessing

Developer Target 1: 73.9% accurate, 1.8× 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 2024326 
(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)))