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

Percentage Accurate: 61.7% → 90.6%

Time: 13.0s

Alternatives: 8

Speedup: 11.3×

• could not determine a ground truth

  1. x = -1.2198457926331911e-151
  2. y = 5.409205432968043e+76
  3. z = 1.4461692344387573e+269
  4. t = 1.6141475829758285e-216

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: 61.7% 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: 90.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3 \cdot 10^{+150}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{expm1}\left(z\right), y, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3e+150)
   (- x (/ (log (fma (expm1 z) y 1.0)) t))
   (- x (/ (* (expm1 z) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3e+150) {
		tmp = x - (log(fma(expm1(z), y, 1.0)) / t);
	} else {
		tmp = x - ((expm1(z) * y) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3e+150)
		tmp = Float64(x - Float64(log(fma(expm1(z), y, 1.0)) / t));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3e+150], N[(x - N[(N[Log[N[(N[(Exp[z] - 1), $MachinePrecision] * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.00000000000000012e150

   1. Initial program 48.3%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-expm1.f64 99.6

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

   5. Applied rewrites 99.6%

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

   if -3.00000000000000012e150 < y

   1. Initial program 66.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 96.4

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

   5. Applied rewrites 96.4%

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

Alternative 2: 87.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(\left(1 - y\right) + y \cdot e^{z}\right) \leq 200:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (log (+ (- 1.0 y) (* y (exp z)))) 200.0)
   (- x (/ (* (expm1 z) y) t))
   (- x (/ (log 1.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (log(((1.0 - y) + (y * exp(z)))) <= 200.0) {
		tmp = x - ((expm1(z) * y) / t);
	} else {
		tmp = x - (log(1.0) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.log(((1.0 - y) + (y * Math.exp(z)))) <= 200.0) {
		tmp = x - ((Math.expm1(z) * y) / t);
	} else {
		tmp = x - (Math.log(1.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.log(((1.0 - y) + (y * math.exp(z)))) <= 200.0:
		tmp = x - ((math.expm1(z) * y) / t)
	else:
		tmp = x - (math.log(1.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) <= 200.0)
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	else
		tmp = Float64(x - Float64(log(1.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 200.0], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < 200

   1. Initial program 62.6%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 94.5

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

   5. Applied rewrites 94.5%

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

   if 200 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))))

   1. Initial program 99.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 51.8%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 87.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\log \left(\left(1 - y\right) + y \cdot e^{z}\right) \leq 200:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (log (+ (- 1.0 y) (* y (exp z)))) 200.0)
   (- x (* (/ (expm1 z) t) y))
   (- x (/ (log 1.0) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (log(((1.0 - y) + (y * exp(z)))) <= 200.0) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(1.0) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.log(((1.0 - y) + (y * Math.exp(z)))) <= 200.0) {
		tmp = x - ((Math.expm1(z) / t) * y);
	} else {
		tmp = x - (Math.log(1.0) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.log(((1.0 - y) + (y * math.exp(z)))) <= 200.0:
		tmp = x - ((math.expm1(z) / t) * y)
	else:
		tmp = x - (math.log(1.0) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) <= 200.0)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(1.0) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision], 200.0], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < 200

   1. Initial program 62.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 94.5

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

   5. Applied rewrites 94.5%

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

   if 200 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))))

   1. Initial program 99.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 51.8%

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 4: 75.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (log(((1.0 - y) + (y * exp(z)))) <= -Inf)
		tmp = x - ((z / t) * y);
	else
		tmp = x - ((y / t) * z);
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) < -inf.0

   1. Initial program 1.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 85.4

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

   5. Applied rewrites 85.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 85.4%

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

   if -inf.0 < (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))))

   1. Initial program 84.2%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. associate-*r* N/A

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

      5. *-commutative N/A

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

      6. +-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 66.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 72.5%

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

Alternative 5: 81.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 2 \cdot 10^{-7}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z, 0.16666666666666666\right), z, 0.5\right) \cdot y, z, y\right) \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 2e-7)
   (- x (/ (log 1.0) t))
   (-
    x
    (/
     (*
      (fma
       (* (fma (fma 0.041666666666666664 z 0.16666666666666666) z 0.5) y)
       z
       y)
      z)
     t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 2e-7) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - ((fma((fma(fma(0.041666666666666664, z, 0.16666666666666666), z, 0.5) * y), z, y) * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 2e-7)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(Float64(fma(Float64(fma(fma(0.041666666666666664, z, 0.16666666666666666), z, 0.5) * y), z, y) * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 2e-7], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(N[(N[(0.041666666666666664 * z + 0.16666666666666666), $MachinePrecision] * z + 0.5), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision] * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 1.9999999999999999e-7

   1. Initial program 81.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 66.0%

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

   if 1.9999999999999999e-7 < (exp.f64 z)

   1. Initial program 57.2%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 94.4

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

   5. Applied rewrites 94.4%

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

   6. Taylor expanded in z around 0

      \[\leadsto x - \frac{z \cdot \color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 94.4%

      \[\leadsto x - \frac{\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.041666666666666664, z \cdot y, 0.16666666666666666 \cdot y\right), z, 0.5 \cdot y\right), z, y\right) \cdot \color{blue}{z}}{t} \]

   9. Taylor expanded in z around 0

      \[\leadsto x - \frac{z \cdot \color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}}{t} \]
   10. Step-by-step derivation

   11. Applied rewrites 94.4%

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

Alternative 6: 86.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.8 \cdot 10^{+151}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right) \cdot y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.8e+151)
   (- x (/ (log (fma z y 1.0)) t))
   (- x (/ (* (expm1 z) y) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.8e+151) {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	} else {
		tmp = x - ((expm1(z) * y) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.8e+151)
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) * y) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.8e+151], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.8e151

   1. Initial program 48.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 58.5

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

   5. Applied rewrites 58.5%

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

   if -3.8e151 < y

   1. Initial program 66.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 96.4

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

   5. Applied rewrites 96.4%

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

Alternative 7: 73.8% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 75.1

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

5. Applied rewrites 75.1%

   \[\leadsto x - \frac{\color{blue}{z \cdot y}}{t} \]
6. Add Preprocessing

Alternative 8: 72.4% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.9%

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

3. Taylor expanded in z around 0

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

   1. *-commutative N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. associate-*r* N/A

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

   5. *-commutative N/A

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

   6. +-commutative N/A

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

   7. lower-*.f64 N/A

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

5. Applied rewrites 65.6%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 73.6%

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

Developer Target 1: 75.1% 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 2024342 
(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)))