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

Percentage Accurate: 61.8% → 98.3%

Time: 19.3s

Alternatives: 11

Speedup: 211.0×

• could not determine a ground truth

  1. x = 3.2718804682883537e+302
  2. y = 5.4359056194432537e+306
  3. z = 1.3416645680160326e+45
  4. t = -8.08222433880183e-71

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: 61.8% 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: 98.3% 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 58.2%

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

   1. remove-double-neg 58.2%

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

   2. neg-mul-1 58.2%

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

   3. *-commutative 58.2%

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

   4. *-commutative 58.2%

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

   5. neg-mul-1 58.2%

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

   6. remove-double-neg 58.2%

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

   7. sub-neg 58.2%

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

   8. associate-+l+ 73.3%

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

   9. cancel-sign-sub 73.3%

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

   10. log1p-def 78.7%

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

   11. cancel-sign-sub 78.7%

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

   12. +-commutative 78.7%

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

   13. unsub-neg 78.7%

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

   14. *-rgt-identity 78.7%

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

   15. distribute-lft-out-- 78.7%

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

   16. expm1-def 96.3%

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

3. Simplified 96.3%

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

5. Final simplification 96.3%

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

Alternative 2: 91.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+131} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -5.4e+131) (not (<= y 1.0)))
   (- x (/ (log1p (* y z)) t))
   (- x (/ y (/ t (expm1 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -5.4e+131) || !(y <= 1.0)) {
		tmp = x - (log1p((y * z)) / t);
	} 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 <= -5.4e+131) || !(y <= 1.0)) {
		tmp = x - (Math.log1p((y * z)) / t);
	} else {
		tmp = x - (y / (t / Math.expm1(z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (y <= -5.4e+131) or not (y <= 1.0):
		tmp = x - (math.log1p((y * z)) / t)
	else:
		tmp = x - (y / (t / math.expm1(z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -5.4e+131) || !(y <= 1.0))
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	else
		tmp = Float64(x - Float64(y / Float64(t / expm1(z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -5.4e+131], N[Not[LessEqual[y, 1.0]], $MachinePrecision]], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $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 < -5.40000000000000008e131 or 1 < y

   1. Initial program 30.5%

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

      1. remove-double-neg 30.5%

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

      2. neg-mul-1 30.5%

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

      3. neg-mul-1 30.5%

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

      4. remove-double-neg 30.5%

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

      5. sub-neg 30.5%

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

      6. associate-+l+ 68.0%

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

      7. cancel-sign-sub 68.0%

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

      8. log1p-def 68.0%

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

      9. cancel-sign-sub 68.0%

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

      10. +-commutative 68.0%

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

      11. neg-mul-1 68.0%

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

      12. *-commutative 68.0%

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

      13. distribute-lft-out 68.0%

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

   3. Simplified 68.0%

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

   5. Taylor expanded in z around 0 82.2%

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

   if -5.40000000000000008e131 < y < 1

   1. Initial program 69.5%

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

      1. remove-double-neg 69.5%

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

      2. neg-mul-1 69.5%

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

      3. *-commutative 69.5%

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

      4. *-commutative 69.5%

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

      5. neg-mul-1 69.5%

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

      6. remove-double-neg 69.5%

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

      7. sub-neg 69.5%

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

      8. associate-+l+ 75.4%

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

      9. cancel-sign-sub 75.4%

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

      10. log1p-def 83.0%

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

      11. cancel-sign-sub 83.0%

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

      12. +-commutative 83.0%

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

      13. unsub-neg 83.0%

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

      14. *-rgt-identity 83.0%

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

      15. distribute-lft-out-- 83.0%

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

      16. expm1-def 96.0%

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

   3. Simplified 96.0%

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

      1. clear-num 95.9%

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

      2. associate-/r/ 95.9%

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

   6. Applied egg-rr 95.9%

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

   7. Taylor expanded in y around 0 79.8%

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

      1. associate-/l* 79.8%

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

      2. expm1-def 95.8%

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

   9. Simplified 95.8%

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

4. Final simplification 91.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+131} \lor \neg \left(y \leq 1\right):\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 86.2% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-70}:\\ \;\;\;\;x - \mathsf{expm1}\left(z\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(\frac{t}{z} + t \cdot -0.5\right) - z \cdot \left(t \cdot -0.08333333333333333\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.2e-70)
   (- x (* (expm1 z) (/ y t)))
   (- x (/ y (- (+ (/ t z) (* t -0.5)) (* z (* t -0.08333333333333333)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.2e-70) {
		tmp = x - (expm1(z) * (y / t));
	} else {
		tmp = x - (y / (((t / z) + (t * -0.5)) - (z * (t * -0.08333333333333333))));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.2e-70) {
		tmp = x - (Math.expm1(z) * (y / t));
	} else {
		tmp = x - (y / (((t / z) + (t * -0.5)) - (z * (t * -0.08333333333333333))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -4.2e-70:
		tmp = x - (math.expm1(z) * (y / t))
	else:
		tmp = x - (y / (((t / z) + (t * -0.5)) - (z * (t * -0.08333333333333333))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.2e-70)
		tmp = Float64(x - Float64(expm1(z) * Float64(y / t)));
	else
		tmp = Float64(x - Float64(y / Float64(Float64(Float64(t / z) + Float64(t * -0.5)) - Float64(z * Float64(t * -0.08333333333333333)))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.2e-70], N[(x - N[(N[(Exp[z] - 1), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(N[(t / z), $MachinePrecision] + N[(t * -0.5), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2000000000000002e-70

   1. Initial program 79.6%

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

      1. remove-double-neg 79.6%

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

      2. neg-mul-1 79.6%

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

      3. *-commutative 79.6%

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

      4. *-commutative 79.6%

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

      5. neg-mul-1 79.6%

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

      6. remove-double-neg 79.6%

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

      7. sub-neg 79.6%

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

      8. associate-+l+ 82.9%

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

      9. cancel-sign-sub 82.9%

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

      10. log1p-def 95.5%

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

      11. cancel-sign-sub 95.5%

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

      12. +-commutative 95.5%

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

      13. unsub-neg 95.5%

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

      14. *-rgt-identity 95.5%

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

      15. distribute-lft-out-- 95.5%

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

      16. 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. Add Preprocessing

   5. Taylor expanded in y around 0 75.9%

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

      1. expm1-def 78.5%

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

      2. associate-/l* 78.4%

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

      3. associate-/r/ 78.5%

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

   7. Simplified 78.5%

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

   if -4.2000000000000002e-70 < z

   1. Initial program 47.1%

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

      1. remove-double-neg 47.1%

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

      2. neg-mul-1 47.1%

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

      3. *-commutative 47.1%

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

      4. *-commutative 47.1%

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

      5. neg-mul-1 47.1%

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

      6. remove-double-neg 47.1%

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

      7. sub-neg 47.1%

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

      8. associate-+l+ 68.2%

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

      9. cancel-sign-sub 68.2%

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

      10. log1p-def 69.8%

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

      11. cancel-sign-sub 69.8%

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

      12. +-commutative 69.8%

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

      13. unsub-neg 69.8%

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

      14. *-rgt-identity 69.8%

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

      15. distribute-lft-out-- 69.8%

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

      16. expm1-def 94.4%

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

   3. Simplified 94.4%

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

      1. clear-num 94.4%

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

      2. associate-/r/ 94.4%

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

   6. Applied egg-rr 94.4%

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

   7. Taylor expanded in y around 0 68.9%

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

      1. associate-/l* 68.9%

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

      2. expm1-def 88.8%

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

   9. Simplified 88.8%

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

   10. Taylor expanded in z around 0 89.0%

       \[\leadsto x - \frac{y}{\color{blue}{-1 \cdot \left(z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)\right) + \left(-0.5 \cdot t + \frac{t}{z}\right)}} \]
   11. Step-by-step derivation

       1. +-commutative 89.0%

          \[\leadsto x - \frac{y}{\color{blue}{\left(-0.5 \cdot t + \frac{t}{z}\right) + -1 \cdot \left(z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)\right)}} \]

       2. mul-1-neg 89.0%

          \[\leadsto x - \frac{y}{\left(-0.5 \cdot t + \frac{t}{z}\right) + \color{blue}{\left(-z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)\right)}} \]

       3. unsub-neg 89.0%

          \[\leadsto x - \frac{y}{\color{blue}{\left(-0.5 \cdot t + \frac{t}{z}\right) - z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)}} \]

       4. *-commutative 89.0%

          \[\leadsto x - \frac{y}{\left(\color{blue}{t \cdot -0.5} + \frac{t}{z}\right) - z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)} \]

       5. fma-def 89.0%

          \[\leadsto x - \frac{y}{\color{blue}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right)} - z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)} \]

       6. distribute-rgt-out 89.0%

          \[\leadsto x - \frac{y}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right) - z \cdot \color{blue}{\left(t \cdot \left(-0.25 + 0.16666666666666666\right)\right)}} \]

       7. metadata-eval 89.0%

          \[\leadsto x - \frac{y}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right) - z \cdot \left(t \cdot \color{blue}{-0.08333333333333333}\right)} \]

   12. Simplified 89.0%

       \[\leadsto x - \frac{y}{\color{blue}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right) - z \cdot \left(t \cdot -0.08333333333333333\right)}} \]
   13. Step-by-step derivation

       1. fma-udef 89.0%

          \[\leadsto x - \frac{y}{\color{blue}{\left(t \cdot -0.5 + \frac{t}{z}\right)} - z \cdot \left(t \cdot -0.08333333333333333\right)} \]

       2. *-commutative 89.0%

          \[\leadsto x - \frac{y}{\left(\color{blue}{-0.5 \cdot t} + \frac{t}{z}\right) - z \cdot \left(t \cdot -0.08333333333333333\right)} \]

       3. +-commutative 89.0%

          \[\leadsto x - \frac{y}{\color{blue}{\left(\frac{t}{z} + -0.5 \cdot t\right)} - z \cdot \left(t \cdot -0.08333333333333333\right)} \]

       4. *-commutative 89.0%

          \[\leadsto x - \frac{y}{\left(\frac{t}{z} + \color{blue}{t \cdot -0.5}\right) - z \cdot \left(t \cdot -0.08333333333333333\right)} \]

   14. Applied egg-rr 89.0%

       \[\leadsto x - \frac{y}{\color{blue}{\left(\frac{t}{z} + t \cdot -0.5\right)} - z \cdot \left(t \cdot -0.08333333333333333\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 85.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{-70}:\\ \;\;\;\;x - \mathsf{expm1}\left(z\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\left(\frac{t}{z} + t \cdot -0.5\right) - z \cdot \left(t \cdot -0.08333333333333333\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.2% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. remove-double-neg 58.2%

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

   2. neg-mul-1 58.2%

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

   3. *-commutative 58.2%

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

   4. *-commutative 58.2%

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

   5. neg-mul-1 58.2%

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

   6. remove-double-neg 58.2%

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

   7. sub-neg 58.2%

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

   8. associate-+l+ 73.3%

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

   9. cancel-sign-sub 73.3%

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

   10. log1p-def 78.7%

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

   11. cancel-sign-sub 78.7%

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

   12. +-commutative 78.7%

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

   13. unsub-neg 78.7%

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

   14. *-rgt-identity 78.7%

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

   15. distribute-lft-out-- 78.7%

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

   16. expm1-def 96.3%

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

3. Simplified 96.3%

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

   1. clear-num 96.3%

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

   2. associate-/r/ 96.3%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in y around 0 71.3%

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

   1. associate-/l* 71.3%

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

   2. expm1-def 85.2%

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

9. Simplified 85.2%

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

10. Final simplification 85.2%

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

Alternative 5: 82.2% accurate, 12.4× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{\left(\frac{t}{z} + t \cdot -0.5\right) - z \cdot \left(t \cdot -0.08333333333333333\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ y (- (+ (/ t z) (* t -0.5)) (* z (* t -0.08333333333333333))))))

C

double code(double x, double y, double z, double t) {
	return x - (y / (((t / z) + (t * -0.5)) - (z * (t * -0.08333333333333333))));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (y / (((t / z) + (t * (-0.5d0))) - (z * (t * (-0.08333333333333333d0)))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (y / (((t / z) + (t * -0.5)) - (z * (t * -0.08333333333333333))));
}

Python

def code(x, y, z, t):
	return x - (y / (((t / z) + (t * -0.5)) - (z * (t * -0.08333333333333333))))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(y / Float64(Float64(Float64(t / z) + Float64(t * -0.5)) - Float64(z * Float64(t * -0.08333333333333333)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (y / (((t / z) + (t * -0.5)) - (z * (t * -0.08333333333333333))));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(y / N[(N[(N[(t / z), $MachinePrecision] + N[(t * -0.5), $MachinePrecision]), $MachinePrecision] - N[(z * N[(t * -0.08333333333333333), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.2%

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

   1. remove-double-neg 58.2%

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

   2. neg-mul-1 58.2%

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

   3. *-commutative 58.2%

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

   4. *-commutative 58.2%

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

   5. neg-mul-1 58.2%

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

   6. remove-double-neg 58.2%

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

   7. sub-neg 58.2%

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

   8. associate-+l+ 73.3%

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

   9. cancel-sign-sub 73.3%

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

   10. log1p-def 78.7%

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

   11. cancel-sign-sub 78.7%

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

   12. +-commutative 78.7%

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

   13. unsub-neg 78.7%

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

   14. *-rgt-identity 78.7%

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

   15. distribute-lft-out-- 78.7%

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

   16. expm1-def 96.3%

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

3. Simplified 96.3%

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

   1. clear-num 96.3%

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

   2. associate-/r/ 96.3%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in y around 0 71.3%

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

   1. associate-/l* 71.3%

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

   2. expm1-def 85.2%

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

9. Simplified 85.2%

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

10. Taylor expanded in z around 0 82.0%

    \[\leadsto x - \frac{y}{\color{blue}{-1 \cdot \left(z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)\right) + \left(-0.5 \cdot t + \frac{t}{z}\right)}} \]
11. Step-by-step derivation

    1. +-commutative 82.0%

       \[\leadsto x - \frac{y}{\color{blue}{\left(-0.5 \cdot t + \frac{t}{z}\right) + -1 \cdot \left(z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)\right)}} \]

    2. mul-1-neg 82.0%

       \[\leadsto x - \frac{y}{\left(-0.5 \cdot t + \frac{t}{z}\right) + \color{blue}{\left(-z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)\right)}} \]

    3. unsub-neg 82.0%

       \[\leadsto x - \frac{y}{\color{blue}{\left(-0.5 \cdot t + \frac{t}{z}\right) - z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)}} \]

    4. *-commutative 82.0%

       \[\leadsto x - \frac{y}{\left(\color{blue}{t \cdot -0.5} + \frac{t}{z}\right) - z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)} \]

    5. fma-def 82.0%

       \[\leadsto x - \frac{y}{\color{blue}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right)} - z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)} \]

    6. distribute-rgt-out 82.0%

       \[\leadsto x - \frac{y}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right) - z \cdot \color{blue}{\left(t \cdot \left(-0.25 + 0.16666666666666666\right)\right)}} \]

    7. metadata-eval 82.0%

       \[\leadsto x - \frac{y}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right) - z \cdot \left(t \cdot \color{blue}{-0.08333333333333333}\right)} \]

12. Simplified 82.0%

    \[\leadsto x - \frac{y}{\color{blue}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right) - z \cdot \left(t \cdot -0.08333333333333333\right)}} \]
13. Step-by-step derivation

    1. fma-udef 82.0%

       \[\leadsto x - \frac{y}{\color{blue}{\left(t \cdot -0.5 + \frac{t}{z}\right)} - z \cdot \left(t \cdot -0.08333333333333333\right)} \]

    2. *-commutative 82.0%

       \[\leadsto x - \frac{y}{\left(\color{blue}{-0.5 \cdot t} + \frac{t}{z}\right) - z \cdot \left(t \cdot -0.08333333333333333\right)} \]

    3. +-commutative 82.0%

       \[\leadsto x - \frac{y}{\color{blue}{\left(\frac{t}{z} + -0.5 \cdot t\right)} - z \cdot \left(t \cdot -0.08333333333333333\right)} \]

    4. *-commutative 82.0%

       \[\leadsto x - \frac{y}{\left(\frac{t}{z} + \color{blue}{t \cdot -0.5}\right) - z \cdot \left(t \cdot -0.08333333333333333\right)} \]

14. Applied egg-rr 82.0%

    \[\leadsto x - \frac{y}{\color{blue}{\left(\frac{t}{z} + t \cdot -0.5\right)} - z \cdot \left(t \cdot -0.08333333333333333\right)} \]

15. Final simplification 82.0%

    \[\leadsto x - \frac{y}{\left(\frac{t}{z} + t \cdot -0.5\right) - z \cdot \left(t \cdot -0.08333333333333333\right)} \]
16. Add Preprocessing

Alternative 6: 81.8% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \frac{1}{-t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9e-51) x (+ x (* y (* z (/ 1.0 (- t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9e-51) {
		tmp = x;
	} else {
		tmp = x + (y * (z * (1.0 / -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 <= (-9d-51)) then
        tmp = x
    else
        tmp = x + (y * (z * (1.0d0 / -t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9e-51) {
		tmp = x;
	} else {
		tmp = x + (y * (z * (1.0 / -t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9e-51:
		tmp = x
	else:
		tmp = x + (y * (z * (1.0 / -t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9e-51)
		tmp = x;
	else
		tmp = Float64(x + Float64(y * Float64(z * Float64(1.0 / Float64(-t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -9e-51)
		tmp = x;
	else
		tmp = x + (y * (z * (1.0 / -t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9e-51], x, N[(x + N[(y * N[(z * N[(1.0 / (-t)), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999948e-51

   1. Initial program 82.7%

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

      1. remove-double-neg 82.7%

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

      2. neg-mul-1 82.7%

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

      3. *-commutative 82.7%

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

      4. *-commutative 82.7%

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

      5. neg-mul-1 82.7%

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

      6. remove-double-neg 82.7%

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

      7. sub-neg 82.7%

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

      8. associate-+l+ 85.1%

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

      9. cancel-sign-sub 85.1%

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

      10. log1p-def 98.7%

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

      11. cancel-sign-sub 98.7%

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

      12. +-commutative 98.7%

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

      13. unsub-neg 98.7%

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

      14. *-rgt-identity 98.7%

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

      15. distribute-lft-out-- 98.7%

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

      16. 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. Add Preprocessing

   5. Taylor expanded in x around inf 67.0%

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

   if -8.99999999999999948e-51 < z

   1. Initial program 46.9%

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

      1. remove-double-neg 46.9%

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

      2. neg-mul-1 46.9%

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

      3. *-commutative 46.9%

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

      4. *-commutative 46.9%

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

      5. neg-mul-1 46.9%

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

      6. remove-double-neg 46.9%

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

      7. sub-neg 46.9%

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

      8. associate-+l+ 67.8%

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

      9. cancel-sign-sub 67.8%

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

      10. log1p-def 69.4%

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

      11. cancel-sign-sub 69.4%

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

      12. +-commutative 69.4%

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

      13. unsub-neg 69.4%

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

      14. *-rgt-identity 69.4%

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

      15. distribute-lft-out-- 69.4%

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

      16. expm1-def 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in z around 0 85.1%

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

      1. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

      1. frac-2neg 88.7%

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

      2. div-inv 88.6%

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

      3. distribute-neg-frac 88.6%

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

   9. Applied egg-rr 88.6%

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

       1. associate-/r/ 89.0%

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

   11. Simplified 89.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \frac{1}{-t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 82.2% accurate, 14.1× speedup

Math

\[\begin{array}{l} \\ x - \frac{y}{t \cdot \left(\frac{1}{z} - \left(z \cdot -0.08333333333333333 + 0.5\right)\right)} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (- x (/ y (* t (- (/ 1.0 z) (+ (* z -0.08333333333333333) 0.5))))))

C

double code(double x, double y, double z, double t) {
	return x - (y / (t * ((1.0 / z) - ((z * -0.08333333333333333) + 0.5))));
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (y / (t * ((1.0d0 / z) - ((z * (-0.08333333333333333d0)) + 0.5d0))))
end function

Java

public static double code(double x, double y, double z, double t) {
	return x - (y / (t * ((1.0 / z) - ((z * -0.08333333333333333) + 0.5))));
}

Python

def code(x, y, z, t):
	return x - (y / (t * ((1.0 / z) - ((z * -0.08333333333333333) + 0.5))))

Julia

function code(x, y, z, t)
	return Float64(x - Float64(y / Float64(t * Float64(Float64(1.0 / z) - Float64(Float64(z * -0.08333333333333333) + 0.5)))))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x - (y / (t * ((1.0 / z) - ((z * -0.08333333333333333) + 0.5))));
end

Wolfram

code[x_, y_, z_, t_] := N[(x - N[(y / N[(t * N[(N[(1.0 / z), $MachinePrecision] - N[(N[(z * -0.08333333333333333), $MachinePrecision] + 0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 58.2%

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

   1. remove-double-neg 58.2%

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

   2. neg-mul-1 58.2%

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

   3. *-commutative 58.2%

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

   4. *-commutative 58.2%

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

   5. neg-mul-1 58.2%

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

   6. remove-double-neg 58.2%

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

   7. sub-neg 58.2%

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

   8. associate-+l+ 73.3%

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

   9. cancel-sign-sub 73.3%

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

   10. log1p-def 78.7%

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

   11. cancel-sign-sub 78.7%

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

   12. +-commutative 78.7%

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

   13. unsub-neg 78.7%

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

   14. *-rgt-identity 78.7%

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

   15. distribute-lft-out-- 78.7%

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

   16. expm1-def 96.3%

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

3. Simplified 96.3%

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

   1. clear-num 96.3%

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

   2. associate-/r/ 96.3%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in y around 0 71.3%

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

   1. associate-/l* 71.3%

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

   2. expm1-def 85.2%

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

9. Simplified 85.2%

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

10. Taylor expanded in z around 0 82.0%

    \[\leadsto x - \frac{y}{\color{blue}{-1 \cdot \left(z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)\right) + \left(-0.5 \cdot t + \frac{t}{z}\right)}} \]
11. Step-by-step derivation

    1. +-commutative 82.0%

       \[\leadsto x - \frac{y}{\color{blue}{\left(-0.5 \cdot t + \frac{t}{z}\right) + -1 \cdot \left(z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)\right)}} \]

    2. mul-1-neg 82.0%

       \[\leadsto x - \frac{y}{\left(-0.5 \cdot t + \frac{t}{z}\right) + \color{blue}{\left(-z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)\right)}} \]

    3. unsub-neg 82.0%

       \[\leadsto x - \frac{y}{\color{blue}{\left(-0.5 \cdot t + \frac{t}{z}\right) - z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)}} \]

    4. *-commutative 82.0%

       \[\leadsto x - \frac{y}{\left(\color{blue}{t \cdot -0.5} + \frac{t}{z}\right) - z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)} \]

    5. fma-def 82.0%

       \[\leadsto x - \frac{y}{\color{blue}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right)} - z \cdot \left(-0.25 \cdot t + 0.16666666666666666 \cdot t\right)} \]

    6. distribute-rgt-out 82.0%

       \[\leadsto x - \frac{y}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right) - z \cdot \color{blue}{\left(t \cdot \left(-0.25 + 0.16666666666666666\right)\right)}} \]

    7. metadata-eval 82.0%

       \[\leadsto x - \frac{y}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right) - z \cdot \left(t \cdot \color{blue}{-0.08333333333333333}\right)} \]

12. Simplified 82.0%

    \[\leadsto x - \frac{y}{\color{blue}{\mathsf{fma}\left(t, -0.5, \frac{t}{z}\right) - z \cdot \left(t \cdot -0.08333333333333333\right)}} \]

13. Taylor expanded in t around 0 81.9%

    \[\leadsto x - \color{blue}{\frac{y}{t \cdot \left(\frac{1}{z} - \left(0.5 + -0.08333333333333333 \cdot z\right)\right)}} \]
14. Step-by-step derivation

    1. *-commutative 81.9%

       \[\leadsto x - \frac{y}{t \cdot \left(\frac{1}{z} - \left(0.5 + \color{blue}{z \cdot -0.08333333333333333}\right)\right)} \]

15. Simplified 81.9%

    \[\leadsto x - \color{blue}{\frac{y}{t \cdot \left(\frac{1}{z} - \left(0.5 + z \cdot -0.08333333333333333\right)\right)}} \]

16. Final simplification 81.9%

    \[\leadsto x - \frac{y}{t \cdot \left(\frac{1}{z} - \left(z \cdot -0.08333333333333333 + 0.5\right)\right)} \]
17. Add Preprocessing

Alternative 8: 78.7% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z -9e-51) x (- x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9e-51) {
		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 <= (-9d-51)) 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 <= -9e-51) {
		tmp = x;
	} else {
		tmp = x - (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9e-51:
		tmp = x
	else:
		tmp = x - (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9e-51)
		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 <= -9e-51)
		tmp = x;
	else
		tmp = x - (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9e-51], x, N[(x - N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999948e-51

   1. Initial program 82.7%

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

      1. remove-double-neg 82.7%

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

      2. neg-mul-1 82.7%

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

      3. *-commutative 82.7%

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

      4. *-commutative 82.7%

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

      5. neg-mul-1 82.7%

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

      6. remove-double-neg 82.7%

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

      7. sub-neg 82.7%

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

      8. associate-+l+ 85.1%

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

      9. cancel-sign-sub 85.1%

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

      10. log1p-def 98.7%

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

      11. cancel-sign-sub 98.7%

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

      12. +-commutative 98.7%

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

      13. unsub-neg 98.7%

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

      14. *-rgt-identity 98.7%

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

      15. distribute-lft-out-- 98.7%

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

      16. 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. Add Preprocessing

   5. Taylor expanded in x around inf 67.0%

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

   if -8.99999999999999948e-51 < z

   1. Initial program 46.9%

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

      1. remove-double-neg 46.9%

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

      2. neg-mul-1 46.9%

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

      3. *-commutative 46.9%

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

      4. *-commutative 46.9%

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

      5. neg-mul-1 46.9%

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

      6. remove-double-neg 46.9%

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

      7. sub-neg 46.9%

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

      8. associate-+l+ 67.8%

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

      9. cancel-sign-sub 67.8%

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

      10. log1p-def 69.4%

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

      11. cancel-sign-sub 69.4%

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

      12. +-commutative 69.4%

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

      13. unsub-neg 69.4%

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

      14. *-rgt-identity 69.4%

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

      15. distribute-lft-out-- 69.4%

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

      16. expm1-def 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in z around 0 85.1%

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

      1. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

      1. associate-/r/ 82.1%

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

   9. Applied egg-rr 82.1%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 81.9% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (if (<= z -9e-51) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9e-51) {
		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 <= (-9d-51)) 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 <= -9e-51) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -9e-51:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9e-51)
		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 <= -9e-51)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9e-51], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -8.99999999999999948e-51

   1. Initial program 82.7%

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

      1. remove-double-neg 82.7%

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

      2. neg-mul-1 82.7%

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

      3. *-commutative 82.7%

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

      4. *-commutative 82.7%

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

      5. neg-mul-1 82.7%

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

      6. remove-double-neg 82.7%

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

      7. sub-neg 82.7%

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

      8. associate-+l+ 85.1%

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

      9. cancel-sign-sub 85.1%

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

      10. log1p-def 98.7%

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

      11. cancel-sign-sub 98.7%

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

      12. +-commutative 98.7%

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

      13. unsub-neg 98.7%

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

      14. *-rgt-identity 98.7%

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

      15. distribute-lft-out-- 98.7%

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

      16. 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. Add Preprocessing

   5. Taylor expanded in x around inf 67.0%

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

   if -8.99999999999999948e-51 < z

   1. Initial program 46.9%

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

      1. remove-double-neg 46.9%

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

      2. neg-mul-1 46.9%

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

      3. *-commutative 46.9%

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

      4. *-commutative 46.9%

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

      5. neg-mul-1 46.9%

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

      6. remove-double-neg 46.9%

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

      7. sub-neg 46.9%

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

      8. associate-+l+ 67.8%

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

      9. cancel-sign-sub 67.8%

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

      10. log1p-def 69.4%

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

      11. cancel-sign-sub 69.4%

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

      12. +-commutative 69.4%

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

      13. unsub-neg 69.4%

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

      14. *-rgt-identity 69.4%

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

      15. distribute-lft-out-- 69.4%

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

      16. expm1-def 94.7%

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

   3. Simplified 94.7%

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

   5. Taylor expanded in z around 0 85.1%

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

      1. associate-/l* 88.7%

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

   7. Simplified 88.7%

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

      1. clear-num 88.6%

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

      2. associate-/r/ 88.6%

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

      3. clear-num 89.0%

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

   9. Applied egg-rr 89.0%

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

4. Final simplification 82.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9 \cdot 10^{-51}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 81.5% accurate, 19.2× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x - (y / ((t / z) + (t * (-0.5d0))))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.2%

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

   1. remove-double-neg 58.2%

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

   2. neg-mul-1 58.2%

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

   3. *-commutative 58.2%

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

   4. *-commutative 58.2%

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

   5. neg-mul-1 58.2%

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

   6. remove-double-neg 58.2%

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

   7. sub-neg 58.2%

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

   8. associate-+l+ 73.3%

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

   9. cancel-sign-sub 73.3%

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

   10. log1p-def 78.7%

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

   11. cancel-sign-sub 78.7%

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

   12. +-commutative 78.7%

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

   13. unsub-neg 78.7%

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

   14. *-rgt-identity 78.7%

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

   15. distribute-lft-out-- 78.7%

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

   16. expm1-def 96.3%

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

3. Simplified 96.3%

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

   1. clear-num 96.3%

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

   2. associate-/r/ 96.3%

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

6. Applied egg-rr 96.3%

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

7. Taylor expanded in y around 0 71.3%

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

   1. associate-/l* 71.3%

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

   2. expm1-def 85.2%

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

9. Simplified 85.2%

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

10. Taylor expanded in z around 0 81.2%

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

11. Final simplification 81.2%

    \[\leadsto x - \frac{y}{\frac{t}{z} + t \cdot -0.5} \]
12. Add Preprocessing

Alternative 11: 72.1% 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 58.2%

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

   1. remove-double-neg 58.2%

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

   2. neg-mul-1 58.2%

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

   3. *-commutative 58.2%

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

   4. *-commutative 58.2%

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

   5. neg-mul-1 58.2%

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

   6. remove-double-neg 58.2%

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

   7. sub-neg 58.2%

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

   8. associate-+l+ 73.3%

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

   9. cancel-sign-sub 73.3%

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

   10. log1p-def 78.7%

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

   11. cancel-sign-sub 78.7%

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

   12. +-commutative 78.7%

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

   13. unsub-neg 78.7%

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

   14. *-rgt-identity 78.7%

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

   15. distribute-lft-out-- 78.7%

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

   16. expm1-def 96.3%

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

3. Simplified 96.3%

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

5. Taylor expanded in x around inf 68.1%

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

6. Final simplification 68.1%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 74.9% 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 2024020 
(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)))