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

Percentage Accurate: 61.5% → 98.4%

Time: 14.0s

Alternatives: 9

Speedup: 211.0×

• could not determine a ground truth

  1. x = -1.0681103143671564e+179
  2. y = 2.658859375117045e-307
  3. z = 5.5413315293193704e+53
  4. t = 1.1555184946441338e+263

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: 98.4% 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 61.4%

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

   1. associate-+l- 78.6%

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

   2. sub-neg 78.6%

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

   3. log1p-def 84.0%

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

   4. neg-sub0 84.0%

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

   5. associate-+l- 84.0%

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

   6. neg-sub0 84.0%

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

   7. neg-mul-1 84.0%

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

   8. *-commutative 84.0%

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

   9. distribute-rgt-out 84.0%

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

   10. +-commutative 84.0%

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

   11. metadata-eval 84.0%

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

   12. sub-neg 84.0%

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

   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: 88.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 61.4%

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

   1. associate-+l- 78.6%

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

   2. sub-neg 78.6%

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

   3. log1p-def 84.0%

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

   4. neg-sub0 84.0%

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

   5. associate-+l- 84.0%

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

   6. neg-sub0 84.0%

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

   7. neg-mul-1 84.0%

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

   8. *-commutative 84.0%

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

   9. distribute-rgt-out 84.0%

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

   10. +-commutative 84.0%

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

   11. metadata-eval 84.0%

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

   12. sub-neg 84.0%

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

   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. inv-pow 99.0%

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

5. Applied egg-rr 99.0%

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

   1. unpow-1 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 78.2%

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

   1. expm1-def 88.9%

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

   2. *-commutative 88.9%

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

   3. associate-/r* 87.5%

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

   4. *-commutative 87.5%

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

10. Simplified 87.5%

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

11. Taylor expanded in t around 0 78.2%

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

    1. *-commutative 78.2%

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

    2. expm1-def 88.8%

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

    3. *-commutative 88.8%

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

13. Simplified 88.8%

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

14. Final simplification 88.8%

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

Alternative 3: 89.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-33} \lor \neg \left(y \leq 3.5 \cdot 10^{-7}\right):\\ \;\;\;\;x + \frac{-1}{\frac{t}{y \cdot z} + t \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.6e-33) (not (<= y 3.5e-7)))
   (+ x (/ -1.0 (+ (/ t (* y z)) (* t 0.5))))
   (- x (* y (/ (expm1 z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.6e-33) || !(y <= 3.5e-7)) {
		tmp = x + (-1.0 / ((t / (y * z)) + (t * 0.5)));
	} 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 <= -2.6e-33) || !(y <= 3.5e-7)) {
		tmp = x + (-1.0 / ((t / (y * z)) + (t * 0.5)));
	} else {
		tmp = x - (y * (Math.expm1(z) / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -2.6e-33) or not (y <= 3.5e-7):
		tmp = x + (-1.0 / ((t / (y * z)) + (t * 0.5)))
	else:
		tmp = x - (y * (math.expm1(z) / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.6e-33) || !(y <= 3.5e-7))
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(t / Float64(y * z)) + Float64(t * 0.5))));
	else
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.6e-33], N[Not[LessEqual[y, 3.5e-7]], $MachinePrecision]], N[(x + N[(-1.0 / N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * 0.5), $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 < -2.59999999999999994e-33 or 3.49999999999999984e-7 < y

   1. Initial program 35.0%

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

      1. associate-+l- 73.6%

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

      2. sub-neg 73.6%

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

      3. log1p-def 73.6%

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

      4. neg-sub0 73.6%

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

      5. associate-+l- 73.6%

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

      6. neg-sub0 73.6%

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

      7. neg-mul-1 73.6%

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

      8. *-commutative 73.6%

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

      9. distribute-rgt-out 73.6%

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

      10. +-commutative 73.6%

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

      11. metadata-eval 73.6%

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

      12. sub-neg 73.6%

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

      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. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 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 60.7%

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

      1. expm1-def 77.1%

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

      2. *-commutative 77.1%

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

      3. associate-/r* 72.1%

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

      4. *-commutative 72.1%

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

   10. Simplified 72.1%

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

   11. Taylor expanded in z around 0 77.1%

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

       1. *-commutative 77.1%

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

   13. Simplified 77.1%

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

   if -2.59999999999999994e-33 < y < 3.49999999999999984e-7

   1. Initial program 82.6%

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

      1. associate-+l- 82.6%

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

      2. sub-neg 82.6%

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

      3. log1p-def 92.4%

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

      4. neg-sub0 92.4%

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

      5. associate-+l- 92.4%

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

      6. neg-sub0 92.4%

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

      7. neg-mul-1 92.4%

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

      8. *-commutative 92.4%

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

      9. distribute-rgt-out 92.3%

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

      10. +-commutative 92.3%

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

      11. metadata-eval 92.3%

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

      12. sub-neg 92.3%

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

      13. expm1-def 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in y around 0 92.2%

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

      1. associate-/l* 92.2%

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

      2. associate-/r/ 92.2%

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

      3. expm1-def 99.8%

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

   6. Simplified 99.8%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.6 \cdot 10^{-33} \lor \neg \left(y \leq 3.5 \cdot 10^{-7}\right):\\ \;\;\;\;x + \frac{-1}{\frac{t}{y \cdot z} + t \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \]

Alternative 4: 89.6% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+16} \lor \neg \left(y \leq 3.5 \cdot 10^{-7}\right):\\ \;\;\;\;x + \frac{-1}{\frac{t}{y \cdot z} + t \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4.6e+16) (not (<= y 3.5e-7)))
   (+ x (/ -1.0 (+ (/ t (* y z)) (* t 0.5))))
   (- x (/ y (/ t (expm1 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.6e+16) || !(y <= 3.5e-7)) {
		tmp = x + (-1.0 / ((t / (y * z)) + (t * 0.5)));
	} else {
		tmp = x - (y / (t / expm1(z)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.6e+16) || !(y <= 3.5e-7)) {
		tmp = x + (-1.0 / ((t / (y * z)) + (t * 0.5)));
	} else {
		tmp = x - (y / (t / Math.expm1(z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -4.6e+16) or not (y <= 3.5e-7):
		tmp = x + (-1.0 / ((t / (y * z)) + (t * 0.5)))
	else:
		tmp = x - (y / (t / math.expm1(z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4.6e+16) || !(y <= 3.5e-7))
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(t / Float64(y * z)) + Float64(t * 0.5))));
	else
		tmp = Float64(x - Float64(y / Float64(t / expm1(z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4.6e+16], N[Not[LessEqual[y, 3.5e-7]], $MachinePrecision]], N[(x + N[(-1.0 / N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6e16 or 3.49999999999999984e-7 < y

   1. Initial program 30.6%

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

      1. associate-+l- 71.9%

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

      2. sub-neg 71.9%

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

      3. log1p-def 71.9%

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

      4. neg-sub0 71.9%

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

      5. associate-+l- 71.9%

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

      6. neg-sub0 71.9%

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

      7. neg-mul-1 71.9%

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

      8. *-commutative 71.9%

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

      9. distribute-rgt-out 71.9%

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

      10. +-commutative 71.9%

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

      11. metadata-eval 71.9%

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

      12. sub-neg 71.9%

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

      13. expm1-def 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
   4. 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. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 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 57.8%

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

      1. expm1-def 74.9%

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

      2. *-commutative 74.9%

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

      3. associate-/r* 69.5%

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

      4. *-commutative 69.5%

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

   10. Simplified 69.5%

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

   11. Taylor expanded in z around 0 74.9%

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

       1. *-commutative 74.9%

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

   13. Simplified 74.9%

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

   if -4.6e16 < y < 3.49999999999999984e-7

   1. Initial program 82.5%

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

      1. associate-+l- 83.2%

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

      2. sub-neg 83.2%

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

      3. log1p-def 92.2%

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

      4. neg-sub0 92.2%

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

      5. associate-+l- 92.2%

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

      6. neg-sub0 92.2%

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

      7. neg-mul-1 92.2%

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

      8. *-commutative 92.2%

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

      9. distribute-rgt-out 92.2%

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

      10. +-commutative 92.2%

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

      11. metadata-eval 92.2%

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

      12. sub-neg 92.2%

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

      13. expm1-def 98.5%

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

   3. Simplified 98.5%

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

   4. Taylor expanded in y around 0 92.1%

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

      1. associate-/l* 92.1%

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

      2. associate-/r/ 92.1%

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

      3. expm1-def 99.8%

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

   6. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. clear-num 99.8%

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

      3. un-div-inv 99.9%

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

   8. Applied egg-rr 99.9%

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

4. Final simplification 89.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.6 \cdot 10^{+16} \lor \neg \left(y \leq 3.5 \cdot 10^{-7}\right):\\ \;\;\;\;x + \frac{-1}{\frac{t}{y \cdot z} + t \cdot 0.5}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \end{array} \]

Alternative 5: 83.9% accurate, 10.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{y \cdot z} + t \cdot 0.5\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{-1}{t_1 - 0.5 \cdot \frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ t (* y z)) (* t 0.5))))
   (if (<= z -2.2e-9)
     (+ x (/ -1.0 (- t_1 (* 0.5 (/ t y)))))
     (+ x (/ -1.0 t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t / (y * z)) + (t * 0.5);
	double tmp;
	if (z <= -2.2e-9) {
		tmp = x + (-1.0 / (t_1 - (0.5 * (t / y))));
	} else {
		tmp = x + (-1.0 / t_1);
	}
	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 = (t / (y * z)) + (t * 0.5d0)
    if (z <= (-2.2d-9)) then
        tmp = x + ((-1.0d0) / (t_1 - (0.5d0 * (t / y))))
    else
        tmp = x + ((-1.0d0) / t_1)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t / (y * z)) + (t * 0.5);
	double tmp;
	if (z <= -2.2e-9) {
		tmp = x + (-1.0 / (t_1 - (0.5 * (t / y))));
	} else {
		tmp = x + (-1.0 / t_1);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t / (y * z)) + (t * 0.5)
	tmp = 0
	if z <= -2.2e-9:
		tmp = x + (-1.0 / (t_1 - (0.5 * (t / y))))
	else:
		tmp = x + (-1.0 / t_1)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t / Float64(y * z)) + Float64(t * 0.5))
	tmp = 0.0
	if (z <= -2.2e-9)
		tmp = Float64(x + Float64(-1.0 / Float64(t_1 - Float64(0.5 * Float64(t / y)))));
	else
		tmp = Float64(x + Float64(-1.0 / t_1));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t / (y * z)) + (t * 0.5);
	tmp = 0.0;
	if (z <= -2.2e-9)
		tmp = x + (-1.0 / (t_1 - (0.5 * (t / y))));
	else
		tmp = x + (-1.0 / t_1);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * 0.5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2.2e-9], N[(x + N[(-1.0 / N[(t$95$1 - N[(0.5 * N[(t / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.1999999999999998e-9

   1. Initial program 82.6%

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

      1. associate-+l- 82.6%

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

      2. sub-neg 82.6%

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

      3. log1p-def 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate-+l- 99.8%

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

      6. neg-sub0 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-rgt-out 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. sub-neg 99.8%

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

      13. expm1-def 99.9%

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

   3. Simplified 99.9%

      \[\leadsto \color{blue}{x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t}} \]
   4. 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. inv-pow 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. unpow-1 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 81.7%

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

      1. expm1-def 81.7%

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

      2. *-commutative 81.7%

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

      3. associate-/r* 81.7%

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

      4. *-commutative 81.7%

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

   10. Simplified 81.7%

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

   11. Taylor expanded in z around 0 67.0%

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

   if -2.1999999999999998e-9 < z

   1. Initial program 51.8%

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

      1. associate-+l- 76.8%

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

      2. sub-neg 76.8%

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

      3. log1p-def 76.8%

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

      4. neg-sub0 76.8%

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

      5. associate-+l- 76.8%

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

      6. neg-sub0 76.8%

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

      7. neg-mul-1 76.8%

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

      8. *-commutative 76.8%

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

      9. distribute-rgt-out 76.8%

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

      10. +-commutative 76.8%

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

      11. metadata-eval 76.8%

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

      12. sub-neg 76.8%

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

      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. Step-by-step derivation

      1. clear-num 98.6%

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

      2. inv-pow 98.6%

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

   5. Applied egg-rr 98.6%

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

      1. unpow-1 98.6%

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

   7. Applied egg-rr 98.6%

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

   8. Taylor expanded in y around 0 76.7%

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

      1. expm1-def 92.1%

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

      2. *-commutative 92.1%

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

      3. associate-/r* 90.2%

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

      4. *-commutative 90.2%

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

   10. Simplified 90.2%

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

   11. Taylor expanded in z around 0 92.1%

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

       1. *-commutative 92.1%

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

   13. Simplified 92.1%

       \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{z \cdot y}} + t \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-9}:\\ \;\;\;\;x + \frac{-1}{\left(\frac{t}{y \cdot z} + t \cdot 0.5\right) - 0.5 \cdot \frac{t}{y}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{y \cdot z} + t \cdot 0.5}\\ \end{array} \]

Alternative 6: 82.3% accurate, 14.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+108}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{y \cdot z} + t \cdot 0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8.2e+108) x (+ x (/ -1.0 (+ (/ t (* y z)) (* t 0.5))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.2e+108) {
		tmp = x;
	} else {
		tmp = x + (-1.0 / ((t / (y * z)) + (t * 0.5)));
	}
	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 <= (-8.2d+108)) then
        tmp = x
    else
        tmp = x + ((-1.0d0) / ((t / (y * z)) + (t * 0.5d0)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.2e+108) {
		tmp = x;
	} else {
		tmp = x + (-1.0 / ((t / (y * z)) + (t * 0.5)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -8.2e+108:
		tmp = x
	else:
		tmp = x + (-1.0 / ((t / (y * z)) + (t * 0.5)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8.2e+108)
		tmp = x;
	else
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(t / Float64(y * z)) + Float64(t * 0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -8.2e+108)
		tmp = x;
	else
		tmp = x + (-1.0 / ((t / (y * z)) + (t * 0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8.2e+108], x, N[(x + N[(-1.0 / N[(N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision] + N[(t * 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.1999999999999998e108

   1. Initial program 84.1%

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

      1. associate-+l- 84.1%

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

      2. sub-neg 84.1%

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

      3. log1p-def 99.8%

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

      4. neg-sub0 99.8%

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

      5. associate-+l- 99.8%

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

      6. neg-sub0 99.8%

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

      7. neg-mul-1 99.8%

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

      8. *-commutative 99.8%

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

      9. distribute-rgt-out 99.8%

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

      10. +-commutative 99.8%

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

      11. metadata-eval 99.8%

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

      12. sub-neg 99.8%

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

      13. expm1-def 99.8%

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

   3. Simplified 99.8%

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

   4. Taylor expanded in z around 0 5.8%

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

      1. *-commutative 5.8%

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

      2. fma-def 6.1%

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

      3. *-commutative 6.1%

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

      4. associate-*l* 6.1%

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

      5. unpow2 6.1%

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

      6. +-commutative 6.1%

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

      7. mul-1-neg 6.1%

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

      8. unsub-neg 6.1%

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

      9. unpow2 6.1%

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

   6. Simplified 6.1%

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

   7. Taylor expanded in y around inf 5.5%

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

      1. associate-/l* 5.5%

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

      2. unpow2 5.5%

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

      3. unpow2 5.5%

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

   9. Simplified 5.5%

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

   10. Taylor expanded in x around inf 62.0%

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

   if -8.1999999999999998e108 < z

   1. Initial program 56.6%

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

      1. associate-+l- 77.4%

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

      2. sub-neg 77.4%

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

      3. log1p-def 80.6%

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

      4. neg-sub0 80.6%

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

      5. associate-+l- 80.6%

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

      6. neg-sub0 80.6%

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

      7. neg-mul-1 80.6%

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

      8. *-commutative 80.6%

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

      9. distribute-rgt-out 80.6%

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

      10. +-commutative 80.6%

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

      11. metadata-eval 80.6%

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

      12. sub-neg 80.6%

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

      13. expm1-def 98.9%

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

   3. Simplified 98.9%

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

      1. clear-num 98.8%

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

      2. inv-pow 98.8%

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

   5. Applied egg-rr 98.8%

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

      1. unpow-1 98.8%

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

   7. Applied egg-rr 98.8%

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

   8. Taylor expanded in y around 0 77.7%

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

      1. expm1-def 90.6%

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

      2. *-commutative 90.6%

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

      3. associate-/r* 89.0%

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

      4. *-commutative 89.0%

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

   10. Simplified 89.0%

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

   11. Taylor expanded in z around 0 87.6%

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

       1. *-commutative 87.6%

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

   13. Simplified 87.6%

       \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{z \cdot y}} + t \cdot 0.5} \]
3. Recombined 2 regimes into one program.

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.2 \cdot 10^{+108}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{y \cdot z} + t \cdot 0.5}\\ \end{array} \]

Alternative 7: 78.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.9e+21) x (- x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.9e+21) {
		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 <= (-1.9d+21)) 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 <= -1.9e+21) {
		tmp = x;
	} else {
		tmp = x - (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.9e+21:
		tmp = x
	else:
		tmp = x - (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.9e+21)
		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 <= -1.9e+21)
		tmp = x;
	else
		tmp = x - (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.9e+21], x, N[(x - N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.9e21

   1. Initial program 82.2%

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

      1. associate-+l- 82.2%

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

      2. sub-neg 82.2%

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

      3. log1p-def 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate-+l- 99.9%

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

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. *-commutative 99.9%

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

      9. distribute-rgt-out 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. sub-neg 99.9%

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

      13. expm1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 21.6%

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

      1. *-commutative 21.6%

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

      2. fma-def 21.8%

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

      3. *-commutative 21.8%

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

      4. associate-*l* 21.8%

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

      5. unpow2 21.8%

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

      6. +-commutative 21.8%

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

      7. mul-1-neg 21.8%

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

      8. unsub-neg 21.8%

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

      9. unpow2 21.8%

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

   6. Simplified 21.8%

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

   7. Taylor expanded in y around inf 21.7%

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

      1. associate-/l* 20.3%

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

      2. unpow2 20.3%

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

      3. unpow2 20.3%

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

   9. Simplified 20.3%

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

   10. Taylor expanded in x around inf 62.1%

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

   if -1.9e21 < z

   1. Initial program 53.2%

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

      1. associate-+l- 77.2%

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

      2. sub-neg 77.2%

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

      3. log1p-def 77.7%

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

      4. neg-sub0 77.7%

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

      5. associate-+l- 77.7%

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

      6. neg-sub0 77.7%

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

      7. neg-mul-1 77.7%

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

      8. *-commutative 77.7%

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

      9. distribute-rgt-out 77.6%

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

      10. +-commutative 77.6%

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

      11. metadata-eval 77.6%

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

      12. sub-neg 77.6%

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

      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 87.4%

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

      1. associate-/l* 88.1%

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

      2. associate-/r/ 85.2%

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

   6. Simplified 85.2%

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

4. Final simplification 78.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \]

Alternative 8: 81.4% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.7e+21) x (- x (/ y (/ t z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.7e+21) {
		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 <= (-3.7d+21)) 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 <= -3.7e+21) {
		tmp = x;
	} else {
		tmp = x - (y / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.7e+21:
		tmp = x
	else:
		tmp = x - (y / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.7e+21)
		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 <= -3.7e+21)
		tmp = x;
	else
		tmp = x - (y / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.7e+21], x, N[(x - N[(y / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.7e21

   1. Initial program 82.2%

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

      1. associate-+l- 82.2%

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

      2. sub-neg 82.2%

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

      3. log1p-def 99.9%

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

      4. neg-sub0 99.9%

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

      5. associate-+l- 99.9%

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

      6. neg-sub0 99.9%

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

      7. neg-mul-1 99.9%

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

      8. *-commutative 99.9%

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

      9. distribute-rgt-out 99.9%

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

      10. +-commutative 99.9%

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

      11. metadata-eval 99.9%

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

      12. sub-neg 99.9%

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

      13. expm1-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in z around 0 21.6%

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

      1. *-commutative 21.6%

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

      2. fma-def 21.8%

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

      3. *-commutative 21.8%

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

      4. associate-*l* 21.8%

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

      5. unpow2 21.8%

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

      6. +-commutative 21.8%

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

      7. mul-1-neg 21.8%

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

      8. unsub-neg 21.8%

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

      9. unpow2 21.8%

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

   6. Simplified 21.8%

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

   7. Taylor expanded in y around inf 21.7%

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

      1. associate-/l* 20.3%

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

      2. unpow2 20.3%

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

      3. unpow2 20.3%

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

   9. Simplified 20.3%

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

   10. Taylor expanded in x around inf 62.1%

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

   if -3.7e21 < z

   1. Initial program 53.2%

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

      1. associate-+l- 77.2%

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

      2. sub-neg 77.2%

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

      3. log1p-def 77.7%

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

      4. neg-sub0 77.7%

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

      5. associate-+l- 77.7%

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

      6. neg-sub0 77.7%

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

      7. neg-mul-1 77.7%

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

      8. *-commutative 77.7%

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

      9. distribute-rgt-out 77.6%

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

      10. +-commutative 77.6%

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

      11. metadata-eval 77.6%

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

      12. sub-neg 77.6%

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

      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. Step-by-step derivation

      1. clear-num 98.7%

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

      2. inv-pow 98.7%

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

   5. Applied egg-rr 98.7%

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

   6. Taylor expanded in z around 0 87.4%

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

      1. associate-/l* 88.1%

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

   8. Simplified 88.1%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.7 \cdot 10^{+21}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z}}\\ \end{array} \]

Alternative 9: 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 61.4%

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

   1. associate-+l- 78.6%

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

   2. sub-neg 78.6%

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

   3. log1p-def 84.0%

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

   4. neg-sub0 84.0%

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

   5. associate-+l- 84.0%

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

   6. neg-sub0 84.0%

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

   7. neg-mul-1 84.0%

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

   8. *-commutative 84.0%

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

   9. distribute-rgt-out 84.0%

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

   10. +-commutative 84.0%

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

   11. metadata-eval 84.0%

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

   12. sub-neg 84.0%

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

   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 61.3%

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

   1. *-commutative 61.3%

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

   2. fma-def 61.3%

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

   3. *-commutative 61.3%

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

   4. associate-*l* 61.3%

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

   5. unpow2 61.3%

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

   6. +-commutative 61.3%

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

   7. mul-1-neg 61.3%

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

   8. unsub-neg 61.3%

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

   9. unpow2 61.3%

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

6. Simplified 61.3%

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

7. Taylor expanded in y around inf 52.6%

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

   1. associate-/l* 52.2%

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

   2. unpow2 52.2%

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

   3. unpow2 52.2%

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

9. Simplified 52.2%

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

10. Taylor expanded in x around inf 72.0%

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

11. Final simplification 72.0%

    \[\leadsto x \]

Developer target: 74.6% 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 2023199 
(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)))