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

Percentage Accurate: 61.3% → 98.5%

Time: 13.8s

Alternatives: 10

Speedup: 211.0×

• could not determine a ground truth

  1. x = 5.4572330358414e+264
  2. y = 1.2531941945704228e+290
  3. z = 1.1542259980161112e+204
  4. t = -8.638272375784938e+238

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: 61.3% 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 58.4%

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

   1. associate-+l- 75.8%

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

   2. sub-neg 75.8%

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

   3. log1p-def 81.6%

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

   4. neg-sub0 81.6%

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

   5. associate-+l- 81.6%

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

   6. neg-sub0 81.6%

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

   7. neg-mul-1 81.6%

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

   8. *-commutative 81.6%

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

   9. distribute-rgt-out 81.6%

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

   10. +-commutative 81.6%

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

   11. metadata-eval 81.6%

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

   12. sub-neg 81.6%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 89.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

   1. associate-+l- 75.8%

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

   2. sub-neg 75.8%

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

   3. log1p-def 81.6%

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

   4. neg-sub0 81.6%

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

   5. associate-+l- 81.6%

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

   6. neg-sub0 81.6%

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

   7. neg-mul-1 81.6%

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

   8. *-commutative 81.6%

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

   9. distribute-rgt-out 81.6%

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

   10. +-commutative 81.6%

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

   11. metadata-eval 81.6%

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

   12. sub-neg 81.6%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 99.0%

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

   2. associate-/r/ 99.0%

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

5. Applied egg-rr 99.0%

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

   1. associate-*l/ 99.0%

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

   2. associate-/l* 99.0%

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

7. Applied egg-rr 99.0%

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

8. Taylor expanded in y around 0 76.4%

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

   1. expm1-def 87.6%

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

   2. *-commutative 87.6%

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

   3. *-commutative 87.6%

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

10. Simplified 87.6%

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

11. Final simplification 87.6%

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

Alternative 3: 90.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

   1. associate-+l- 75.8%

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

   2. sub-neg 75.8%

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

   3. log1p-def 81.6%

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

   4. neg-sub0 81.6%

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

   5. associate-+l- 81.6%

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

   6. neg-sub0 81.6%

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

   7. neg-mul-1 81.6%

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

   8. *-commutative 81.6%

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

   9. distribute-rgt-out 81.6%

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

   10. +-commutative 81.6%

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

   11. metadata-eval 81.6%

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

   12. sub-neg 81.6%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 99.0%

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

   2. associate-/r/ 99.0%

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

5. Applied egg-rr 99.0%

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

   1. associate-*l/ 99.0%

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

   2. associate-/l* 99.0%

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

7. Applied egg-rr 99.0%

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

8. Taylor expanded in y around 0 76.4%

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

   1. expm1-def 87.6%

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

   2. *-commutative 87.6%

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

   3. *-commutative 87.6%

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

10. Simplified 87.6%

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

11. Taylor expanded in t around 0 76.4%

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

    1. associate-/r* 76.4%

       \[\leadsto x - \frac{1}{\color{blue}{\frac{\frac{t}{e^{z} - 1}}{y}} + t \cdot 0.5} \]

    2. expm1-def 88.1%

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

13. Simplified 88.1%

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

14. Final simplification 88.1%

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

Alternative 4: 89.3% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{-1}{t \cdot 0.5 + \frac{t}{y \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -6.6e+19)
   (+ x (/ -1.0 (+ (* t 0.5) (/ t (* y z)))))
   (- x (* y (/ (expm1 z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.6e+19) {
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * z))));
	} else {
		tmp = x - (y * (expm1(z) / t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -6.6e+19) {
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * z))));
	} else {
		tmp = x - (y * (Math.expm1(z) / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -6.6e+19:
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * z))))
	else:
		tmp = x - (y * (math.expm1(z) / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -6.6e+19)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(t * 0.5) + Float64(t / Float64(y * z)))));
	else
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -6.6e+19], N[(x + N[(-1.0 / N[(N[(t * 0.5), $MachinePrecision] + N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -6.6e19

   1. Initial program 42.9%

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

      1. associate-+l- 74.2%

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

      2. sub-neg 74.2%

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

      3. log1p-def 74.2%

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

      4. neg-sub0 74.2%

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

      5. associate-+l- 74.2%

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

      6. neg-sub0 74.2%

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

      7. neg-mul-1 74.2%

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

      8. *-commutative 74.2%

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

      9. distribute-rgt-out 74.2%

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

      10. +-commutative 74.2%

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

      11. metadata-eval 74.2%

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

      12. sub-neg 74.2%

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

      13. expm1-def 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-*l/ 99.9%

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

      2. associate-/l* 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in y around 0 54.8%

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

      1. expm1-def 67.6%

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

      2. *-commutative 67.6%

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

      3. *-commutative 67.6%

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

   10. Simplified 67.6%

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

   11. Taylor expanded in z around 0 67.6%

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

       1. *-commutative 67.6%

          \[\leadsto x - \frac{1}{\frac{t}{\color{blue}{z \cdot y}} + t \cdot 0.5} \]

   13. Simplified 67.6%

       \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{z \cdot y}} + t \cdot 0.5} \]

   if -6.6e19 < y

   1. Initial program 63.1%

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

      1. associate-+l- 76.3%

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

      2. sub-neg 76.3%

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

      3. log1p-def 83.8%

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

      4. neg-sub0 83.8%

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

      5. associate-+l- 83.8%

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

      6. neg-sub0 83.8%

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

      7. neg-mul-1 83.8%

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

      8. *-commutative 83.8%

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

      9. distribute-rgt-out 83.8%

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

      10. +-commutative 83.8%

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

      11. metadata-eval 83.8%

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

      12. sub-neg 83.8%

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

      13. expm1-def 98.8%

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

   3. Simplified 98.8%

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

   4. Taylor expanded in y around 0 83.3%

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

      1. associate-/l* 82.4%

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

      2. associate-/r/ 83.3%

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

      3. expm1-def 94.0%

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

   6. Simplified 94.0%

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

4. Final simplification 87.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+19}:\\ \;\;\;\;x + \frac{-1}{t \cdot 0.5 + \frac{t}{y \cdot z}}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \]

Alternative 5: 84.2% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-194}:\\ \;\;\;\;x + \frac{-1}{\left(t \cdot 0.5 + \frac{t}{y \cdot z}\right) - 0.5 \cdot \frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\left(y \cdot 0.5\right) \cdot \left(z \cdot z\right)}{t} + y \cdot \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1e-194)
   (+ x (/ -1.0 (- (+ (* t 0.5) (/ t (* y z))) (* 0.5 (/ t y)))))
   (- x (+ (/ (* (* y 0.5) (* z z)) t) (* y (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1e-194) {
		tmp = x + (-1.0 / (((t * 0.5) + (t / (y * z))) - (0.5 * (t / y))));
	} else {
		tmp = x - ((((y * 0.5) * (z * z)) / t) + (y * (z / t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1d-194)) then
        tmp = x + ((-1.0d0) / (((t * 0.5d0) + (t / (y * z))) - (0.5d0 * (t / y))))
    else
        tmp = x - ((((y * 0.5d0) * (z * z)) / t) + (y * (z / t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1e-194) {
		tmp = x + (-1.0 / (((t * 0.5) + (t / (y * z))) - (0.5 * (t / y))));
	} else {
		tmp = x - ((((y * 0.5) * (z * z)) / t) + (y * (z / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1e-194:
		tmp = x + (-1.0 / (((t * 0.5) + (t / (y * z))) - (0.5 * (t / y))))
	else:
		tmp = x - ((((y * 0.5) * (z * z)) / t) + (y * (z / t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1e-194)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(Float64(t * 0.5) + Float64(t / Float64(y * z))) - Float64(0.5 * Float64(t / y)))));
	else
		tmp = Float64(x - Float64(Float64(Float64(Float64(y * 0.5) * Float64(z * z)) / t) + Float64(y * Float64(z / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1e-194)
		tmp = x + (-1.0 / (((t * 0.5) + (t / (y * z))) - (0.5 * (t / y))));
	else
		tmp = x - ((((y * 0.5) * (z * z)) / t) + (y * (z / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1e-194], N[(x + N[(-1.0 / N[(N[(N[(t * 0.5), $MachinePrecision] + N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(0.5 * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(y * 0.5), $MachinePrecision] * N[(z * z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.00000000000000002e-194

   1. Initial program 64.3%

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

      1. associate-+l- 75.4%

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

      2. sub-neg 75.4%

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

      3. log1p-def 87.3%

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

      4. neg-sub0 87.3%

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

      5. associate-+l- 87.3%

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

      6. neg-sub0 87.3%

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

      7. neg-mul-1 87.3%

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

      8. *-commutative 87.3%

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

      9. distribute-rgt-out 87.3%

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

      10. +-commutative 87.3%

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

      11. metadata-eval 87.3%

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

      12. sub-neg 87.3%

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

      13. expm1-def 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-*l/ 99.9%

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

      2. associate-/l* 99.8%

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

   7. Applied egg-rr 99.8%

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

   8. Taylor expanded in y around 0 78.0%

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

      1. expm1-def 83.5%

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

      2. *-commutative 83.5%

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

      3. *-commutative 83.5%

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

   10. Simplified 83.5%

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

   11. Taylor expanded in z around 0 73.4%

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

   if -1.00000000000000002e-194 < z

   1. Initial program 53.0%

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

      1. associate-+l- 76.2%

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

      2. sub-neg 76.2%

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

      3. log1p-def 76.2%

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

      4. neg-sub0 76.2%

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

      5. associate-+l- 76.2%

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

      6. neg-sub0 76.2%

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

      7. neg-mul-1 76.2%

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

      8. *-commutative 76.2%

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

      9. distribute-rgt-out 76.2%

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

      10. +-commutative 76.2%

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

      11. metadata-eval 76.2%

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

      12. sub-neg 76.2%

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

      13. expm1-def 98.2%

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

   3. Simplified 98.2%

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

   4. Taylor expanded in y around 0 75.6%

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

      1. expm1-def 90.9%

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

   6. Simplified 90.9%

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

   7. Taylor expanded in z around 0 90.9%

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

      1. associate-*r/ 91.9%

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

      2. +-commutative 91.9%

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

      3. associate-*r/ 91.9%

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

      4. associate-*r* 91.9%

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

      5. unpow2 91.9%

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

   9. Simplified 91.9%

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

4. Final simplification 83.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-194}:\\ \;\;\;\;x + \frac{-1}{\left(t \cdot 0.5 + \frac{t}{y \cdot z}\right) - 0.5 \cdot \frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\frac{\left(y \cdot 0.5\right) \cdot \left(z \cdot z\right)}{t} + y \cdot \frac{z}{t}\right)\\ \end{array} \]

Alternative 6: 81.0% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ x + \frac{-1}{t \cdot 0.5 + \frac{t}{y \cdot z}} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + (-1.0 / ((t * 0.5) + (t / (y * 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 + ((-1.0d0) / ((t * 0.5d0) + (t / (y * z))))
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp = code(x, y, z, t)
	tmp = x + (-1.0 / ((t * 0.5) + (t / (y * z))));
end

Wolfram

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

Derivation

1. Initial program 58.4%

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

   1. associate-+l- 75.8%

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

   2. sub-neg 75.8%

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

   3. log1p-def 81.6%

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

   4. neg-sub0 81.6%

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

   5. associate-+l- 81.6%

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

   6. neg-sub0 81.6%

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

   7. neg-mul-1 81.6%

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

   8. *-commutative 81.6%

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

   9. distribute-rgt-out 81.6%

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

   10. +-commutative 81.6%

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

   11. metadata-eval 81.6%

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

   12. sub-neg 81.6%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 99.0%

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

   2. associate-/r/ 99.0%

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

5. Applied egg-rr 99.0%

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

   1. associate-*l/ 99.0%

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

   2. associate-/l* 99.0%

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

7. Applied egg-rr 99.0%

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

8. Taylor expanded in y around 0 76.4%

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

   1. expm1-def 87.6%

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

   2. *-commutative 87.6%

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

   3. *-commutative 87.6%

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

10. Simplified 87.6%

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

11. Taylor expanded in z around 0 80.3%

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

    1. *-commutative 80.3%

       \[\leadsto x - \frac{1}{\frac{t}{\color{blue}{z \cdot y}} + t \cdot 0.5} \]

13. Simplified 80.3%

    \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{z \cdot y}} + t \cdot 0.5} \]

14. Final simplification 80.3%

    \[\leadsto x + \frac{-1}{t \cdot 0.5 + \frac{t}{y \cdot z}} \]

Alternative 7: 78.6% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.45e+20) x (- x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.45e+20) {
		tmp = x;
	} else {
		tmp = x - (z * (y / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.45d+20)) then
        tmp = x
    else
        tmp = x - (z * (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.45e+20) {
		tmp = x;
	} else {
		tmp = x - (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.45e+20:
		tmp = x
	else:
		tmp = x - (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.45e+20)
		tmp = x;
	else
		tmp = Float64(x - Float64(z * Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.45e+20)
		tmp = x;
	else
		tmp = x - (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.45e+20], x, N[(x - N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.45e20

   1. Initial program 80.4%

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

      1. associate-+l- 80.4%

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

      2. sub-neg 80.4%

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

      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 z around 0 11.8%

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

      1. fma-def 11.8%

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

      2. associate-/l* 11.4%

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

      3. associate-/r/ 8.8%

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

      4. unpow2 8.8%

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

      5. +-commutative 8.8%

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

      6. mul-1-neg 8.8%

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

      7. unsub-neg 8.8%

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

      8. unpow2 8.8%

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

      9. associate-/l* 8.6%

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

      10. associate-/r/ 8.9%

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

   6. Simplified 8.9%

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

   7. Taylor expanded in z around inf 12.1%

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

      1. unpow2 12.1%

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

      2. *-commutative 12.1%

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

      3. associate-/l* 11.7%

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

      4. unpow2 11.7%

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

      5. associate-/l* 24.7%

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

      6. associate-*r/ 24.7%

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

      7. associate-/r* 21.8%

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

      8. *-commutative 21.8%

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

      9. associate-/r* 21.8%

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

   9. Simplified 21.8%

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

   10. Taylor expanded in y around 0 12.2%

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

       1. associate-*r/ 12.2%

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

       2. associate-*r* 12.2%

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

       3. unpow2 12.2%

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

   12. Simplified 12.2%

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

   13. Taylor expanded in x around inf 62.2%

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

   if -1.45e20 < z

   1. Initial program 50.3%

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

      1. associate-+l- 74.2%

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

      2. sub-neg 74.2%

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

      3. log1p-def 74.8%

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

      4. neg-sub0 74.8%

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

      5. associate-+l- 74.8%

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

      6. neg-sub0 74.8%

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

      7. neg-mul-1 74.8%

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

      8. *-commutative 74.8%

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

      9. distribute-rgt-out 74.8%

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

      10. +-commutative 74.8%

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

      11. metadata-eval 74.8%

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

      12. sub-neg 74.8%

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

      13. expm1-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in z around 0 86.1%

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

      1. associate-/l* 87.3%

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

      2. associate-/r/ 81.4%

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

   6. Simplified 81.4%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.45 \cdot 10^{+20}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 8: 82.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.25e+16) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.25e+16) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.25d+16)) then
        tmp = x
    else
        tmp = x - (y * (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.25e+16) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.25e+16:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.25e+16)
		tmp = x;
	else
		tmp = Float64(x - Float64(y * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.25e+16)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.25e+16], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25e16

   1. Initial program 80.4%

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

      1. associate-+l- 80.4%

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

      2. sub-neg 80.4%

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

      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 z around 0 11.8%

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

      1. fma-def 11.8%

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

      2. associate-/l* 11.4%

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

      3. associate-/r/ 8.8%

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

      4. unpow2 8.8%

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

      5. +-commutative 8.8%

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

      6. mul-1-neg 8.8%

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

      7. unsub-neg 8.8%

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

      8. unpow2 8.8%

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

      9. associate-/l* 8.6%

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

      10. associate-/r/ 8.9%

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

   6. Simplified 8.9%

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

   7. Taylor expanded in z around inf 12.1%

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

      1. unpow2 12.1%

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

      2. *-commutative 12.1%

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

      3. associate-/l* 11.7%

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

      4. unpow2 11.7%

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

      5. associate-/l* 24.7%

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

      6. associate-*r/ 24.7%

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

      7. associate-/r* 21.8%

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

      8. *-commutative 21.8%

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

      9. associate-/r* 21.8%

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

   9. Simplified 21.8%

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

   10. Taylor expanded in y around 0 12.2%

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

       1. associate-*r/ 12.2%

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

       2. associate-*r* 12.2%

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

       3. unpow2 12.2%

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

   12. Simplified 12.2%

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

   13. Taylor expanded in x around inf 62.2%

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

   if -1.25e16 < z

   1. Initial program 50.3%

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

      1. associate-+l- 74.2%

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

      2. sub-neg 74.2%

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

      3. log1p-def 74.8%

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

      4. neg-sub0 74.8%

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

      5. associate-+l- 74.8%

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

      6. neg-sub0 74.8%

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

      7. neg-mul-1 74.8%

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

      8. *-commutative 74.8%

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

      9. distribute-rgt-out 74.8%

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

      10. +-commutative 74.8%

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

      11. metadata-eval 74.8%

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

      12. sub-neg 74.8%

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

      13. expm1-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in y around 0 74.0%

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

      1. associate-/l* 71.8%

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

      2. associate-/r/ 74.0%

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

      3. expm1-def 87.7%

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

   6. Simplified 87.7%

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

   7. Taylor expanded in z around 0 87.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{+16}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 9: 82.0% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z -7e+18) x (- x (/ y (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7e+18) {
		tmp = x;
	} else {
		tmp = x - (y / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-7d+18)) then
        tmp = x
    else
        tmp = x - (y / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7e+18) {
		tmp = x;
	} else {
		tmp = x - (y / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7e+18:
		tmp = x
	else:
		tmp = x - (y / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7e+18)
		tmp = x;
	else
		tmp = Float64(x - Float64(y / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7e+18)
		tmp = x;
	else
		tmp = x - (y / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7e+18], x, N[(x - N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7e18

   1. Initial program 80.4%

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

      1. associate-+l- 80.4%

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

      2. sub-neg 80.4%

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

      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 z around 0 11.8%

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

      1. fma-def 11.8%

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

      2. associate-/l* 11.4%

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

      3. associate-/r/ 8.8%

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

      4. unpow2 8.8%

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

      5. +-commutative 8.8%

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

      6. mul-1-neg 8.8%

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

      7. unsub-neg 8.8%

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

      8. unpow2 8.8%

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

      9. associate-/l* 8.6%

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

      10. associate-/r/ 8.9%

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

   6. Simplified 8.9%

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

   7. Taylor expanded in z around inf 12.1%

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

      1. unpow2 12.1%

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

      2. *-commutative 12.1%

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

      3. associate-/l* 11.7%

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

      4. unpow2 11.7%

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

      5. associate-/l* 24.7%

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

      6. associate-*r/ 24.7%

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

      7. associate-/r* 21.8%

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

      8. *-commutative 21.8%

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

      9. associate-/r* 21.8%

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

   9. Simplified 21.8%

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

   10. Taylor expanded in y around 0 12.2%

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

       1. associate-*r/ 12.2%

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

       2. associate-*r* 12.2%

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

       3. unpow2 12.2%

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

   12. Simplified 12.2%

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

   13. Taylor expanded in x around inf 62.2%

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

   if -7e18 < z

   1. Initial program 50.3%

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

      1. associate-+l- 74.2%

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

      2. sub-neg 74.2%

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

      3. log1p-def 74.8%

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

      4. neg-sub0 74.8%

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

      5. associate-+l- 74.8%

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

      6. neg-sub0 74.8%

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

      7. neg-mul-1 74.8%

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

      8. *-commutative 74.8%

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

      9. distribute-rgt-out 74.8%

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

      10. +-commutative 74.8%

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

      11. metadata-eval 74.8%

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

      12. sub-neg 74.8%

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

      13. expm1-def 98.7%

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

   3. Simplified 98.7%

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

   4. Taylor expanded in z around 0 86.1%

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

      1. associate-/l* 87.3%

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

      2. associate-/r/ 81.4%

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

   6. Simplified 81.4%

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

      1. associate-*l/ 86.1%

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

      2. associate-/l* 87.3%

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

   8. Applied egg-rr 87.3%

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

4. Final simplification 80.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7 \cdot 10^{+18}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 10: 70.9% accurate, 211.0× speedup

Math

\[\begin{array}{l} \\ x \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 x)

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x
end function

Java

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

Python

def code(x, y, z, t):
	return x

Julia

function code(x, y, z, t)
	return x
end

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

Derivation

1. Initial program 58.4%

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

   1. associate-+l- 75.8%

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

   2. sub-neg 75.8%

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

   3. log1p-def 81.6%

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

   4. neg-sub0 81.6%

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

   5. associate-+l- 81.6%

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

   6. neg-sub0 81.6%

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

   7. neg-mul-1 81.6%

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

   8. *-commutative 81.6%

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

   9. distribute-rgt-out 81.6%

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

   10. +-commutative 81.6%

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

   11. metadata-eval 81.6%

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

   12. sub-neg 81.6%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 57.7%

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

   1. fma-def 57.7%

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

   2. associate-/l* 52.4%

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

   3. associate-/r/ 56.9%

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

   4. unpow2 56.9%

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

   5. +-commutative 56.9%

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

   6. mul-1-neg 56.9%

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

   7. unsub-neg 56.9%

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

   8. unpow2 56.9%

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

   9. associate-/l* 57.4%

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

   10. associate-/r/ 55.3%

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

6. Simplified 55.3%

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

7. Taylor expanded in z around inf 49.2%

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

   1. unpow2 49.2%

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

   2. *-commutative 49.2%

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

   3. associate-/l* 47.1%

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

   4. unpow2 47.1%

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

   5. associate-/l* 50.9%

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

   6. associate-*r/ 50.9%

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

   7. associate-/r* 52.0%

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

   8. *-commutative 52.0%

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

   9. associate-/r* 52.0%

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

9. Simplified 52.0%

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

10. Taylor expanded in y around 0 56.3%

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

    1. associate-*r/ 56.3%

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

    2. associate-*r* 56.3%

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

    3. unpow2 56.3%

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

12. Simplified 56.3%

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

13. Taylor expanded in x around inf 69.9%

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

14. Final simplification 69.9%

    \[\leadsto x \]

Developer target: 74.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5}{y \cdot t}\\ \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{t_1}{z \cdot z}\right) - t_1 \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- 0.5) (* y t))))
   (if (< z -2.8874623088207947e+119)
     (- (- x (/ t_1 (* z z))) (* t_1 (/ (/ 2.0 z) (* z z))))
     (- x (/ (log (+ 1.0 (* z y))) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -0.5d0 / (y * t)
    if (z < (-2.8874623088207947d+119)) then
        tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0d0 / z) / (z * z)))
    else
        tmp = x - (log((1.0d0 + (z * y))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (Math.log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -0.5 / (y * t)
	tmp = 0
	if z < -2.8874623088207947e+119:
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)))
	else:
		tmp = x - (math.log((1.0 + (z * y))) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5) / Float64(y * t))
	tmp = 0.0
	if (z < -2.8874623088207947e+119)
		tmp = Float64(Float64(x - Float64(t_1 / Float64(z * z))) - Float64(t_1 * Float64(Float64(2.0 / z) / Float64(z * z))));
	else
		tmp = Float64(x - Float64(log(Float64(1.0 + Float64(z * y))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -0.5 / (y * t);
	tmp = 0.0;
	if (z < -2.8874623088207947e+119)
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	else
		tmp = x - (log((1.0 + (z * y))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-0.5) / N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.8874623088207947e+119], N[(N[(x - N[(t$95$1 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[(2.0 / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(1.0 + N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023238 
(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)))