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

Percentage Accurate: 61.5% → 98.5%

Time: 14.7s

Alternatives: 7

Speedup: 30.1×

• could not determine a ground truth

  1. x = 1.2396357949716825e-33
  2. y = 1.3947470714642663e-308
  3. z = 1.4959382933519313e+231
  4. t = -2.708844357862378e+168

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: 61.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. associate-+l- 73.7%

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

   2. sub-neg 73.7%

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

   3. log1p-def 79.0%

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

   4. neg-sub0 79.0%

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

   5. associate-+l- 79.0%

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

   6. neg-sub0 79.0%

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

   7. neg-mul-1 79.0%

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

   8. *-commutative 79.0%

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

   9. distribute-rgt-out 79.0%

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

   10. +-commutative 79.0%

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

   11. metadata-eval 79.0%

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

   12. sub-neg 79.0%

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

   13. expm1-def 97.6%

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

3. Simplified 97.6%

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

4. Final simplification 97.6%

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

Alternative 2: 85.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-81}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{z}{t} \cdot -0.5\right) - \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4e-81)
   (- x (/ (expm1 z) (/ t y)))
   (+ x (* y (- (* z (* (/ z t) -0.5)) (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4e-81) {
		tmp = x - (expm1(z) / (t / y));
	} else {
		tmp = x + (y * ((z * ((z / t) * -0.5)) - (z / t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4e-81) {
		tmp = x - (Math.expm1(z) / (t / y));
	} else {
		tmp = x + (y * ((z * ((z / t) * -0.5)) - (z / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4e-81:
		tmp = x - (math.expm1(z) / (t / y))
	else:
		tmp = x + (y * ((z * ((z / t) * -0.5)) - (z / t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4e-81)
		tmp = Float64(x - Float64(expm1(z) / Float64(t / y)));
	else
		tmp = Float64(x + Float64(y * Float64(Float64(z * Float64(Float64(z / t) * -0.5)) - Float64(z / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4e-81], N[(x - N[(N[(Exp[z] - 1), $MachinePrecision] / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(z * N[(N[(z / t), $MachinePrecision] * -0.5), $MachinePrecision]), $MachinePrecision] - N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.9999999999999998e-81

   1. Initial program 79.4%

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

      1. associate-+l- 80.5%

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

      2. sub-neg 80.5%

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

      3. log1p-def 91.7%

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

      4. neg-sub0 91.7%

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

      5. associate-+l- 91.7%

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

      6. neg-sub0 91.7%

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

      7. neg-mul-1 91.7%

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

      8. *-commutative 91.7%

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

      9. distribute-rgt-out 91.8%

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

      10. +-commutative 91.8%

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

      11. metadata-eval 91.8%

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

      12. sub-neg 91.8%

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

      13. expm1-def 99.1%

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

   3. Simplified 99.1%

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

   4. Taylor expanded in y around 0 73.3%

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

      1. expm1-def 76.7%

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

      2. associate-/l* 77.5%

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

   6. Simplified 77.5%

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

   if -3.9999999999999998e-81 < z

   1. Initial program 50.6%

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

      1. associate-+l- 69.2%

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

      2. sub-neg 69.2%

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

      3. log1p-def 70.5%

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

      4. neg-sub0 70.5%

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

      5. associate-+l- 70.5%

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

      6. neg-sub0 70.5%

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

      7. neg-mul-1 70.5%

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

      8. *-commutative 70.5%

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

      9. distribute-rgt-out 70.5%

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

      10. +-commutative 70.5%

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

      11. metadata-eval 70.5%

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

      12. sub-neg 70.5%

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

      13. expm1-def 96.6%

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

   3. Simplified 96.6%

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

   4. Taylor expanded in y around 0 70.2%

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

      1. expm1-def 86.5%

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

      2. associate-/l* 84.4%

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

   6. Simplified 84.4%

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

   7. Taylor expanded in z around 0 86.8%

      \[\leadsto x - \color{blue}{\left(\frac{y \cdot z}{t} + 0.5 \cdot \frac{y \cdot {z}^{2}}{t}\right)} \]
   8. Step-by-step derivation

      1. *-commutative 86.8%

         \[\leadsto x - \left(\frac{\color{blue}{z \cdot y}}{t} + 0.5 \cdot \frac{y \cdot {z}^{2}}{t}\right) \]

      2. associate-/l* 86.8%

         \[\leadsto x - \left(\frac{z \cdot y}{t} + 0.5 \cdot \color{blue}{\frac{y}{\frac{t}{{z}^{2}}}}\right) \]

      3. unpow2 86.8%

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

   9. Simplified 86.8%

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

   10. Taylor expanded in y around -inf 88.6%

       \[\leadsto x - \color{blue}{-1 \cdot \left(y \cdot \left(-0.5 \cdot \frac{{z}^{2}}{t} + -1 \cdot \frac{z}{t}\right)\right)} \]
   11. Step-by-step derivation

       1. mul-1-neg 88.6%

          \[\leadsto x - \color{blue}{\left(-y \cdot \left(-0.5 \cdot \frac{{z}^{2}}{t} + -1 \cdot \frac{z}{t}\right)\right)} \]

       2. distribute-rgt-neg-in 88.6%

          \[\leadsto x - \color{blue}{y \cdot \left(-\left(-0.5 \cdot \frac{{z}^{2}}{t} + -1 \cdot \frac{z}{t}\right)\right)} \]

       3. mul-1-neg 88.6%

          \[\leadsto x - y \cdot \left(-\left(-0.5 \cdot \frac{{z}^{2}}{t} + \color{blue}{\left(-\frac{z}{t}\right)}\right)\right) \]

       4. unsub-neg 88.6%

          \[\leadsto x - y \cdot \left(-\color{blue}{\left(-0.5 \cdot \frac{{z}^{2}}{t} - \frac{z}{t}\right)}\right) \]

       5. *-commutative 88.6%

          \[\leadsto x - y \cdot \left(-\left(\color{blue}{\frac{{z}^{2}}{t} \cdot -0.5} - \frac{z}{t}\right)\right) \]

       6. unpow2 88.6%

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

       7. associate-*r/ 88.6%

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

       8. associate-*l* 88.6%

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

   12. Simplified 88.6%

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

4. Final simplification 84.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-81}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{z}{t} \cdot -0.5\right) - \frac{z}{t}\right)\\ \end{array} \]

Alternative 3: 90.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+22}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.05e+22) (- x (/ (expm1 z) (/ t y))) (- x (/ (log1p (* y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.05e+22) {
		tmp = x - (expm1(z) / (t / y));
	} else {
		tmp = x - (log1p((y * z)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.05e+22) {
		tmp = x - (Math.expm1(z) / (t / y));
	} else {
		tmp = x - (Math.log1p((y * z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.05e+22:
		tmp = x - (math.expm1(z) / (t / y))
	else:
		tmp = x - (math.log1p((y * z)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.05e+22)
		tmp = Float64(x - Float64(expm1(z) / Float64(t / y)));
	else
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.05e+22], N[(x - N[(N[(Exp[z] - 1), $MachinePrecision] / N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0499999999999999e22

   1. Initial program 85.4%

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

      1. associate-+l- 85.4%

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

      2. sub-neg 85.4%

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

      3. log1p-def 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. neg-sub0 100.0%

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

      7. neg-mul-1 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 78.9%

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

      1. expm1-def 78.9%

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

      2. associate-/l* 78.8%

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

   6. Simplified 78.8%

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

   if -1.0499999999999999e22 < z

   1. Initial program 53.1%

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

      1. remove-double-neg 53.1%

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

      2. sub0-neg 53.1%

         \[\leadsto x - \color{blue}{\left(0 - \left(-\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)\right)} \]

      3. sub0-neg 53.1%

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

      4. remove-double-neg 53.1%

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

      5. sub-neg 53.1%

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

      6. associate-+l+ 69.2%

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

      7. log1p-def 70.8%

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

      8. +-commutative 70.8%

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

      9. unsub-neg 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in z around 0 70.5%

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

   5. Taylor expanded in y around 0 95.9%

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

      1. *-commutative 95.9%

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

   7. Simplified 95.9%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+22}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{\frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \]

Alternative 4: 90.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+22}:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.05e+22) (- x (/ (* y (expm1 z)) t)) (- x (/ (log1p (* y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.05e+22) {
		tmp = x - ((y * expm1(z)) / t);
	} else {
		tmp = x - (log1p((y * z)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.05e+22) {
		tmp = x - ((y * Math.expm1(z)) / t);
	} else {
		tmp = x - (Math.log1p((y * z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.05e+22:
		tmp = x - ((y * math.expm1(z)) / t)
	else:
		tmp = x - (math.log1p((y * z)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.05e+22)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	else
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.05e+22], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.0499999999999999e22

   1. Initial program 85.4%

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

      1. associate-+l- 85.4%

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

      2. sub-neg 85.4%

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

      3. log1p-def 100.0%

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

      4. neg-sub0 100.0%

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

      5. associate-+l- 100.0%

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

      6. neg-sub0 100.0%

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

      7. neg-mul-1 100.0%

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

      8. *-commutative 100.0%

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

      9. distribute-rgt-out 100.0%

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

      10. +-commutative 100.0%

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

      11. metadata-eval 100.0%

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

      12. sub-neg 100.0%

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

      13. expm1-def 100.0%

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

   3. Simplified 100.0%

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

   4. Taylor expanded in y around 0 78.9%

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

      1. expm1-def 78.9%

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

      2. *-commutative 78.9%

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

   6. Simplified 78.9%

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

   if -1.0499999999999999e22 < z

   1. Initial program 53.1%

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

      1. remove-double-neg 53.1%

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

      2. sub0-neg 53.1%

         \[\leadsto x - \color{blue}{\left(0 - \left(-\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)\right)} \]

      3. sub0-neg 53.1%

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

      4. remove-double-neg 53.1%

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

      5. sub-neg 53.1%

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

      6. associate-+l+ 69.2%

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

      7. log1p-def 70.8%

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

      8. +-commutative 70.8%

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

      9. unsub-neg 70.8%

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

   3. Simplified 70.8%

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

   4. Taylor expanded in z around 0 70.5%

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

   5. Taylor expanded in y around 0 95.9%

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

      1. *-commutative 95.9%

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

   7. Simplified 95.9%

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

4. Final simplification 91.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{+22}:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \]

Alternative 5: 71.2% accurate, 30.1× speedup

Math

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

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(z * Float64(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[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

   1. associate-+l- 73.7%

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

   2. sub-neg 73.7%

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

   3. log1p-def 79.0%

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

   4. neg-sub0 79.0%

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

   5. associate-+l- 79.0%

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

   6. neg-sub0 79.0%

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

   7. neg-mul-1 79.0%

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

   8. *-commutative 79.0%

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

   9. distribute-rgt-out 79.0%

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

   10. +-commutative 79.0%

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

   11. metadata-eval 79.0%

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

   12. sub-neg 79.0%

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

   13. expm1-def 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in z around 0 72.5%

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

   1. associate-/l* 73.9%

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

   2. associate-/r/ 72.4%

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

6. Simplified 72.4%

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

7. Final simplification 72.4%

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

Alternative 6: 73.5% accurate, 30.1× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.2%

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

   1. associate-+l- 73.7%

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

   2. sub-neg 73.7%

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

   3. log1p-def 79.0%

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

   4. neg-sub0 79.0%

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

   5. associate-+l- 79.0%

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

   6. neg-sub0 79.0%

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

   7. neg-mul-1 79.0%

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

   8. *-commutative 79.0%

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

   9. distribute-rgt-out 79.0%

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

   10. +-commutative 79.0%

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

   11. metadata-eval 79.0%

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

   12. sub-neg 79.0%

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

   13. expm1-def 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in z around 0 72.5%

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

   1. associate-/l* 73.9%

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

   2. associate-/r/ 72.4%

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

6. Simplified 72.4%

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

   1. expm1-log1p-u 67.6%

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

   2. expm1-udef 62.4%

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

   3. *-commutative 62.4%

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

   4. clear-num 62.4%

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

   5. un-div-inv 62.4%

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

8. Applied egg-rr 62.4%

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

   1. expm1-def 67.6%

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

   2. expm1-log1p 72.4%

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

   3. associate-/r/ 73.9%

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

10. Simplified 73.9%

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

11. Final simplification 73.9%

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

Alternative 7: 73.6% accurate, 30.1× speedup

Math

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

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(y / Float64(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[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 62.2%

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

   1. associate-+l- 73.7%

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

   2. sub-neg 73.7%

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

   3. log1p-def 79.0%

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

   4. neg-sub0 79.0%

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

   5. associate-+l- 79.0%

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

   6. neg-sub0 79.0%

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

   7. neg-mul-1 79.0%

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

   8. *-commutative 79.0%

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

   9. distribute-rgt-out 79.0%

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

   10. +-commutative 79.0%

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

   11. metadata-eval 79.0%

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

   12. sub-neg 79.0%

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

   13. expm1-def 97.6%

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

3. Simplified 97.6%

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

4. Taylor expanded in z around 0 72.5%

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

   1. associate-/l* 73.9%

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

6. Simplified 73.9%

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

7. Final simplification 73.9%

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

Developer target: 73.6% 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 2023196 
(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)))