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

Percentage Accurate: 61.8% → 98.3%

Time: 21.8s

Alternatives: 7

Speedup: 211.0×

• could not determine a ground truth

  1. x = -5.89673992738072e-305
  2. y = 4.082416222321767e+157
  3. z = 4.2637775213496745e+250
  4. t = -1.5636021403018136e+193

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 58.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

   6. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

   7. neg-mul-1 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

   9. distribute-rgt-out N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

   14. expm1-define N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

   15. expm1-lowering-expm1.f64 97.3%

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

3. Simplified 97.3%

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

Alternative 2: 93.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{-1}{\frac{t \cdot \left(y \cdot 0.5\right) + \frac{t}{\mathsf{expm1}\left(z\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.1e-29)
   (+ x (/ -1.0 (/ (+ (* t (* y 0.5)) (/ t (expm1 z))) y)))
   (- x (/ (log1p (* y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.1e-29) {
		tmp = x + (-1.0 / (((t * (y * 0.5)) + (t / expm1(z))) / y));
	} else {
		tmp = x - (log1p((y * z)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.1e-29) {
		tmp = x + (-1.0 / (((t * (y * 0.5)) + (t / Math.expm1(z))) / y));
	} else {
		tmp = x - (Math.log1p((y * z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.1e-29:
		tmp = x + (-1.0 / (((t * (y * 0.5)) + (t / math.expm1(z))) / y))
	else:
		tmp = x - (math.log1p((y * z)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.1e-29)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(Float64(t * Float64(y * 0.5)) + Float64(t / expm1(z))) / y)));
	else
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.1e-29], N[(x + N[(-1.0 / N[(N[(N[(t * N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.10000000000000026e-29

   1. Initial program 84.6%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      6. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

      15. expm1-lowering-expm1.f64 99.3%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

   3. Simplified 99.3%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{t}{\log \left(1 + y \cdot \left(e^{z} - 1\right)\right)}}}\right)\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{t}{\log \left(1 + y \cdot \left(e^{z} - 1\right)\right)}\right)}\right)\right) \]

      3. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \color{blue}{\log \left(1 + y \cdot \left(e^{z} - 1\right)\right)}\right)\right)\right) \]

      4. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \left(\mathsf{log1p}\left(y \cdot \left(e^{z} - 1\right)\right)\right)\right)\right)\right) \]

      5. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{log1p.f64}\left(\left(y \cdot \left(e^{z} - 1\right)\right)\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right)\right)\right)\right) \]

      7. expm1-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      8. expm1-lowering-expm1.f64 99.3%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(t, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.3%

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

   7. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \color{blue}{\left(\frac{\frac{1}{2} \cdot \left(t \cdot y\right) + \frac{t}{e^{z} - 1}}{y}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\left(\frac{1}{2} \cdot \left(t \cdot y\right) + \frac{t}{e^{z} - 1}\right), \color{blue}{y}\right)\right)\right) \]

      2. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{2} \cdot \left(t \cdot y\right)\right), \left(\frac{t}{e^{z} - 1}\right)\right), y\right)\right)\right) \]

      3. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\left(t \cdot y\right) \cdot \frac{1}{2}\right), \left(\frac{t}{e^{z} - 1}\right)\right), y\right)\right)\right) \]

      4. associate-*l* N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(y \cdot \frac{1}{2}\right)\right), \left(\frac{t}{e^{z} - 1}\right)\right), y\right)\right)\right) \]

      5. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(t \cdot \left(\frac{1}{2} \cdot y\right)\right), \left(\frac{t}{e^{z} - 1}\right)\right), y\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(t, \left(\frac{1}{2} \cdot y\right)\right), \left(\frac{t}{e^{z} - 1}\right)\right), y\right)\right)\right) \]

      7. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(t, \left(y \cdot \frac{1}{2}\right)\right), \left(\frac{t}{e^{z} - 1}\right)\right), y\right)\right)\right) \]

      8. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right), \left(\frac{t}{e^{z} - 1}\right)\right), y\right)\right)\right) \]

      9. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(t, \left(e^{z} - 1\right)\right)\right), y\right)\right)\right) \]

      10. expm1-define N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(t, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), y\right)\right)\right) \]

      11. expm1-lowering-expm1.f64 84.9%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(1, \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{*.f64}\left(t, \mathsf{*.f64}\left(y, \frac{1}{2}\right)\right), \mathsf{/.f64}\left(t, \mathsf{expm1.f64}\left(z\right)\right)\right), y\right)\right)\right) \]

   9. Simplified 84.9%

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

   if -3.10000000000000026e-29 < z

   1. Initial program 46.0%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      6. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

      15. expm1-lowering-expm1.f64 96.5%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

   3. Simplified 96.5%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\color{blue}{\left(y \cdot z\right)}\right), t\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 96.4%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, z\right)\right), t\right)\right) \]

   7. Simplified 96.4%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{-29}:\\ \;\;\;\;x + \frac{-1}{\frac{t \cdot \left(y \cdot 0.5\right) + \frac{t}{\mathsf{expm1}\left(z\right)}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.6% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{if}\;y \leq -6.1 \cdot 10^{+53}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.45 \cdot 10^{-40}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log1p (* y z)) t))))
   (if (<= y -6.1e+53)
     t_1
     (if (<= y 3.45e-40) (- x (* y (/ (expm1 z) t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log1p((y * z)) / t);
	double tmp;
	if (y <= -6.1e+53) {
		tmp = t_1;
	} else if (y <= 3.45e-40) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = x - (Math.log1p((y * z)) / t);
	double tmp;
	if (y <= -6.1e+53) {
		tmp = t_1;
	} else if (y <= 3.45e-40) {
		tmp = x - (y * (Math.expm1(z) / t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x - (math.log1p((y * z)) / t)
	tmp = 0
	if y <= -6.1e+53:
		tmp = t_1
	elif y <= 3.45e-40:
		tmp = x - (y * (math.expm1(z) / t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log1p(Float64(y * z)) / t))
	tmp = 0.0
	if (y <= -6.1e+53)
		tmp = t_1;
	elseif (y <= 3.45e-40)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -6.1e+53], t$95$1, If[LessEqual[y, 3.45e-40], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -6.1000000000000002e53 or 3.4499999999999998e-40 < y

   1. Initial program 27.1%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      6. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

      15. expm1-lowering-expm1.f64 98.8%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

   3. Simplified 98.8%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\color{blue}{\left(y \cdot z\right)}\right), t\right)\right) \]
   6. Step-by-step derivation

      1. *-lowering-*.f64 87.4%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, z\right)\right), t\right)\right) \]

   7. Simplified 87.4%

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

   if -6.1000000000000002e53 < y < 3.4499999999999998e-40

   1. Initial program 77.6%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      6. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

      15. expm1-lowering-expm1.f64 96.4%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

   3. Simplified 96.4%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y \cdot \left(e^{z} - 1\right)}{t}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(y \cdot \color{blue}{\frac{e^{z} - 1}{t}}\right)\right) \]

      2. div-sub N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(y \cdot \left(\frac{e^{z}}{t} - \color{blue}{\frac{1}{t}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{e^{z} - 1}{\color{blue}{t}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(e^{z} - 1\right), \color{blue}{t}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(z\right)\right), t\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 96.8%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(z\right), t\right)\right)\right) \]

   7. Simplified 96.8%

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

Alternative 4: 86.7% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

   6. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

   7. neg-mul-1 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

   9. distribute-rgt-out N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

   14. expm1-define N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

   15. expm1-lowering-expm1.f64 97.3%

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

3. Simplified 97.3%

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

5. Taylor expanded in y around 0

   \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y \cdot \left(e^{z} - 1\right)}{t}\right)}\right) \]
6. Step-by-step derivation

   1. associate-/l* N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(y \cdot \color{blue}{\frac{e^{z} - 1}{t}}\right)\right) \]

   2. div-sub N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \left(y \cdot \left(\frac{e^{z}}{t} - \color{blue}{\frac{1}{t}}\right)\right)\right) \]

   3. *-lowering-*.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)}\right)\right) \]

   4. div-sub N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{e^{z} - 1}{\color{blue}{t}}\right)\right)\right) \]

   5. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(e^{z} - 1\right), \color{blue}{t}\right)\right)\right) \]

   6. expm1-define N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(z\right)\right), t\right)\right)\right) \]

   7. expm1-lowering-expm1.f64 85.5%

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(z\right), t\right)\right)\right) \]

7. Simplified 85.5%

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

Alternative 5: 81.2% accurate, 7.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-22}:\\ \;\;\;\;x + \frac{\frac{1}{t}}{\frac{\frac{-1}{y} + -0.5 \cdot \frac{z \cdot \left(y \cdot y - y\right)}{y \cdot y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{-1}{t} - \frac{z \cdot 0.5}{t}\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.75e-22)
   (+
    x
    (/
     (/ 1.0 t)
     (/ (+ (/ -1.0 y) (* -0.5 (/ (* z (- (* y y) y)) (* y y)))) z)))
   (+ x (* y (* z (- (/ -1.0 t) (/ (* z 0.5) t)))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.75e-22) {
		tmp = x + ((1.0 / t) / (((-1.0 / y) + (-0.5 * ((z * ((y * y) - y)) / (y * y)))) / z));
	} else {
		tmp = x + (y * (z * ((-1.0 / t) - ((z * 0.5) / 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 <= (-1.75d-22)) then
        tmp = x + ((1.0d0 / t) / ((((-1.0d0) / y) + ((-0.5d0) * ((z * ((y * y) - y)) / (y * y)))) / z))
    else
        tmp = x + (y * (z * (((-1.0d0) / t) - ((z * 0.5d0) / t))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.75e-22) {
		tmp = x + ((1.0 / t) / (((-1.0 / y) + (-0.5 * ((z * ((y * y) - y)) / (y * y)))) / z));
	} else {
		tmp = x + (y * (z * ((-1.0 / t) - ((z * 0.5) / t))));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.75e-22:
		tmp = x + ((1.0 / t) / (((-1.0 / y) + (-0.5 * ((z * ((y * y) - y)) / (y * y)))) / z))
	else:
		tmp = x + (y * (z * ((-1.0 / t) - ((z * 0.5) / t))))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.75e-22)
		tmp = Float64(x + Float64(Float64(1.0 / t) / Float64(Float64(Float64(-1.0 / y) + Float64(-0.5 * Float64(Float64(z * Float64(Float64(y * y) - y)) / Float64(y * y)))) / z)));
	else
		tmp = Float64(x + Float64(y * Float64(z * Float64(Float64(-1.0 / t) - Float64(Float64(z * 0.5) / t)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.75e-22)
		tmp = x + ((1.0 / t) / (((-1.0 / y) + (-0.5 * ((z * ((y * y) - y)) / (y * y)))) / z));
	else
		tmp = x + (y * (z * ((-1.0 / t) - ((z * 0.5) / t))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.75e-22], N[(x + N[(N[(1.0 / t), $MachinePrecision] / N[(N[(N[(-1.0 / y), $MachinePrecision] + N[(-0.5 * N[(N[(z * N[(N[(y * y), $MachinePrecision] - y), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(z * N[(N[(-1.0 / t), $MachinePrecision] - N[(N[(z * 0.5), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.75000000000000003e-22

   1. Initial program 84.6%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      6. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

      15. expm1-lowering-expm1.f64 99.9%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

   3. Simplified 99.9%

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

      1. clear-num N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{1}{\color{blue}{\frac{t}{\log \left(1 + y \cdot \left(e^{z} - 1\right)\right)}}}\right)\right) \]

      2. div-inv N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{1}{t \cdot \color{blue}{\frac{1}{\log \left(1 + y \cdot \left(e^{z} - 1\right)\right)}}}\right)\right) \]

      3. associate-/r* N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(\frac{\frac{1}{t}}{\color{blue}{\frac{1}{\log \left(1 + y \cdot \left(e^{z} - 1\right)\right)}}}\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\frac{1}{t}\right), \color{blue}{\left(\frac{1}{\log \left(1 + y \cdot \left(e^{z} - 1\right)\right)}\right)}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \left(\frac{\color{blue}{1}}{\log \left(1 + y \cdot \left(e^{z} - 1\right)\right)}\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(1, \color{blue}{\log \left(1 + y \cdot \left(e^{z} - 1\right)\right)}\right)\right)\right) \]

      7. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(1, \left(\mathsf{log1p}\left(y \cdot \left(e^{z} - 1\right)\right)\right)\right)\right)\right) \]

      8. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(1, \mathsf{log1p.f64}\left(\left(y \cdot \left(e^{z} - 1\right)\right)\right)\right)\right)\right) \]

      9. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right)\right)\right)\right) \]

      10. expm1-define N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right)\right)\right)\right) \]

      11. expm1-lowering-expm1.f64 99.7%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(1, \mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right)\right)\right)\right) \]

   6. Applied egg-rr 99.7%

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

   7. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \color{blue}{\left(\frac{\frac{-1}{2} \cdot \frac{z \cdot \left(y + -1 \cdot {y}^{2}\right)}{{y}^{2}} + \frac{1}{y}}{z}\right)}\right)\right) \]
   8. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\left(\frac{-1}{2} \cdot \frac{z \cdot \left(y + -1 \cdot {y}^{2}\right)}{{y}^{2}} + \frac{1}{y}\right), \color{blue}{z}\right)\right)\right) \]

      2. +-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\left(\frac{1}{y} + \frac{-1}{2} \cdot \frac{z \cdot \left(y + -1 \cdot {y}^{2}\right)}{{y}^{2}}\right), z\right)\right)\right) \]

      3. +-lowering-+.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\left(\frac{1}{y}\right), \left(\frac{-1}{2} \cdot \frac{z \cdot \left(y + -1 \cdot {y}^{2}\right)}{{y}^{2}}\right)\right), z\right)\right)\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \left(\frac{-1}{2} \cdot \frac{z \cdot \left(y + -1 \cdot {y}^{2}\right)}{{y}^{2}}\right)\right), z\right)\right)\right) \]

      5. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(\frac{-1}{2}, \left(\frac{z \cdot \left(y + -1 \cdot {y}^{2}\right)}{{y}^{2}}\right)\right)\right), z\right)\right)\right) \]

      6. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\left(z \cdot \left(y + -1 \cdot {y}^{2}\right)\right), \left({y}^{2}\right)\right)\right)\right), z\right)\right)\right) \]

      7. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(y + -1 \cdot {y}^{2}\right)\right), \left({y}^{2}\right)\right)\right)\right), z\right)\right)\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(y + \left(\mathsf{neg}\left({y}^{2}\right)\right)\right)\right), \left({y}^{2}\right)\right)\right)\right), z\right)\right)\right) \]

      9. unsub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \left(y - {y}^{2}\right)\right), \left({y}^{2}\right)\right)\right)\right), z\right)\right)\right) \]

      10. --lowering--.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \left({y}^{2}\right)\right)\right), \left({y}^{2}\right)\right)\right)\right), z\right)\right)\right) \]

      11. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \left(y \cdot y\right)\right)\right), \left({y}^{2}\right)\right)\right)\right), z\right)\right)\right) \]

      12. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \left({y}^{2}\right)\right)\right)\right), z\right)\right)\right) \]

      13. unpow2 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \left(y \cdot y\right)\right)\right)\right), z\right)\right)\right) \]

      14. *-lowering-*.f64 68.1%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{/.f64}\left(1, t\right), \mathsf{/.f64}\left(\mathsf{+.f64}\left(\mathsf{/.f64}\left(1, y\right), \mathsf{*.f64}\left(\frac{-1}{2}, \mathsf{/.f64}\left(\mathsf{*.f64}\left(z, \mathsf{\_.f64}\left(y, \mathsf{*.f64}\left(y, y\right)\right)\right), \mathsf{*.f64}\left(y, y\right)\right)\right)\right), z\right)\right)\right) \]

   9. Simplified 68.1%

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

   if -1.75000000000000003e-22 < z

   1. Initial program 46.6%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      6. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

      15. expm1-lowering-expm1.f64 96.3%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

   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 y around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y \cdot \left(e^{z} - 1\right)}{t}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(y \cdot \color{blue}{\frac{e^{z} - 1}{t}}\right)\right) \]

      2. div-sub N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(y \cdot \left(\frac{e^{z}}{t} - \color{blue}{\frac{1}{t}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{e^{z} - 1}{\color{blue}{t}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(e^{z} - 1\right), \color{blue}{t}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(z\right)\right), t\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 88.9%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(z\right), t\right)\right)\right) \]

   7. Simplified 88.9%

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

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(z \cdot \left(\frac{1}{2} \cdot \frac{z}{t} + \frac{1}{t}\right)\right)}\right)\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(z \cdot \left(\frac{\frac{1}{2} \cdot z}{t} + \frac{\color{blue}{1}}{t}\right)\right)\right)\right) \]

      2. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(z \cdot \left(\frac{z \cdot \frac{1}{2}}{t} + \frac{1}{t}\right)\right)\right)\right) \]

      3. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(z \cdot \left(z \cdot \frac{\frac{1}{2}}{t} + \frac{\color{blue}{1}}{t}\right)\right)\right)\right) \]

      4. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(z \cdot \left(z \cdot \frac{\frac{1}{2} \cdot 1}{t} + \frac{1}{t}\right)\right)\right)\right) \]

      5. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(z \cdot \left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{t}\right) + \frac{1}{t}\right)\right)\right)\right) \]

      6. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \color{blue}{\left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{t}\right) + \frac{1}{t}\right)}\right)\right)\right) \]

      7. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(z \cdot \frac{\frac{1}{2} \cdot 1}{t} + \frac{1}{t}\right)\right)\right)\right) \]

      8. metadata-eval N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(z \cdot \frac{\frac{1}{2}}{t} + \frac{1}{t}\right)\right)\right)\right) \]

      9. associate-*r/ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(\frac{z \cdot \frac{1}{2}}{t} + \frac{\color{blue}{1}}{t}\right)\right)\right)\right) \]

      10. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(\frac{\frac{1}{2} \cdot z}{t} + \frac{1}{t}\right)\right)\right)\right) \]

      11. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(\frac{1}{2} \cdot \frac{z}{t} + \frac{\color{blue}{1}}{t}\right)\right)\right)\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(\frac{1}{t} + \color{blue}{\frac{1}{2} \cdot \frac{z}{t}}\right)\right)\right)\right) \]

      13. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(\frac{1}{t} + \frac{\frac{1}{2} \cdot z}{\color{blue}{t}}\right)\right)\right)\right) \]

      14. *-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(\frac{1}{t} + \frac{z \cdot \frac{1}{2}}{t}\right)\right)\right)\right) \]

      15. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(\frac{1}{t} + z \cdot \color{blue}{\frac{\frac{1}{2}}{t}}\right)\right)\right)\right) \]

      16. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(\frac{1}{t} + z \cdot \frac{\frac{1}{2} \cdot 1}{t}\right)\right)\right)\right) \]

      17. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \left(\frac{1}{t} + z \cdot \left(\frac{1}{2} \cdot \color{blue}{\frac{1}{t}}\right)\right)\right)\right)\right) \]

      18. +-lowering-+.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\left(\frac{1}{t}\right), \color{blue}{\left(z \cdot \left(\frac{1}{2} \cdot \frac{1}{t}\right)\right)}\right)\right)\right)\right) \]

      19. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, t\right), \left(\color{blue}{z} \cdot \left(\frac{1}{2} \cdot \frac{1}{t}\right)\right)\right)\right)\right)\right) \]

      20. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, t\right), \left(z \cdot \frac{\frac{1}{2} \cdot 1}{\color{blue}{t}}\right)\right)\right)\right)\right) \]

      21. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, t\right), \left(z \cdot \frac{\frac{1}{2}}{t}\right)\right)\right)\right)\right) \]

      22. associate-*r/ N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{*.f64}\left(z, \mathsf{+.f64}\left(\mathsf{/.f64}\left(1, t\right), \left(\frac{z \cdot \frac{1}{2}}{\color{blue}{t}}\right)\right)\right)\right)\right) \]

   10. Simplified 88.8%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.75 \cdot 10^{-22}:\\ \;\;\;\;x + \frac{\frac{1}{t}}{\frac{\frac{-1}{y} + -0.5 \cdot \frac{z \cdot \left(y \cdot y - y\right)}{y \cdot y}}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \left(\frac{-1}{t} - \frac{z \cdot 0.5}{t}\right)\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 82.4% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+53}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7.6e+53) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.6e+53) {
		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 <= (-7.6d+53)) 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 <= -7.6e+53) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7.6e+53:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7.6e+53)
		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 <= -7.6e+53)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7.6e+53], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.59999999999999995e53

   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. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      6. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

      15. expm1-lowering-expm1.f64 99.8%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

   3. Simplified 99.8%

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

   5. Taylor expanded in x around inf

      \[\leadsto \color{blue}{x} \]
   6. Step-by-step derivation

   7. Simplified 66.1%

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

   if -7.59999999999999995e53 < z

   1. Initial program 51.7%

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

      1. --lowering--.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

      3. sub-neg N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

      4. associate-+l+ N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      5. log1p-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      6. log1p-lowering-log1p.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

      7. neg-mul-1 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

      8. *-commutative N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

      9. distribute-rgt-out N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      10. *-lowering-*.f64 N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

      11. +-commutative N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

      12. metadata-eval N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

      13. sub-neg N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

      14. expm1-define N/A

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

      15. expm1-lowering-expm1.f64 96.7%

          \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

   3. Simplified 96.7%

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

   5. Taylor expanded in y around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{y \cdot \left(e^{z} - 1\right)}{t}\right)}\right) \]
   6. Step-by-step derivation

      1. associate-/l* N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(y \cdot \color{blue}{\frac{e^{z} - 1}{t}}\right)\right) \]

      2. div-sub N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \left(y \cdot \left(\frac{e^{z}}{t} - \color{blue}{\frac{1}{t}}\right)\right)\right) \]

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{e^{z}}{t} - \frac{1}{t}\right)}\right)\right) \]

      4. div-sub N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \left(\frac{e^{z} - 1}{\color{blue}{t}}\right)\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(e^{z} - 1\right), \color{blue}{t}\right)\right)\right) \]

      6. expm1-define N/A

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\left(\mathsf{expm1}\left(z\right)\right), t\right)\right)\right) \]

      7. expm1-lowering-expm1.f64 87.3%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(\mathsf{expm1.f64}\left(z\right), t\right)\right)\right) \]

   7. Simplified 87.3%

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

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \color{blue}{\left(\frac{z}{t}\right)}\right)\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 86.8%

         \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{*.f64}\left(y, \mathsf{/.f64}\left(z, \color{blue}{t}\right)\right)\right) \]

   10. Simplified 86.8%

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

Alternative 7: 72.5% 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 58.0%

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

   1. --lowering--.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \color{blue}{\left(\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\right)}\right) \]

   2. /-lowering-/.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 - y\right) + y \cdot e^{z}\right), \color{blue}{t}\right)\right) \]

   3. sub-neg N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(\left(1 + \left(\mathsf{neg}\left(y\right)\right)\right) + y \cdot e^{z}\right), t\right)\right) \]

   4. associate-+l+ N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\log \left(1 + \left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

   5. log1p-define N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\left(\mathsf{log1p}\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

   6. log1p-lowering-log1p.f64 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(\left(\mathsf{neg}\left(y\right)\right) + y \cdot e^{z}\right)\right), t\right)\right) \]

   7. neg-mul-1 N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + y \cdot e^{z}\right)\right), t\right)\right) \]

   8. *-commutative N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(-1 \cdot y + e^{z} \cdot y\right)\right), t\right)\right) \]

   9. distribute-rgt-out N/A

      \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\left(y \cdot \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

   10. *-lowering-*.f64 N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(-1 + e^{z}\right)\right)\right), t\right)\right) \]

   11. +-commutative N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + -1\right)\right)\right), t\right)\right) \]

   12. metadata-eval N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} + \left(\mathsf{neg}\left(1\right)\right)\right)\right)\right), t\right)\right) \]

   13. sub-neg N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(e^{z} - 1\right)\right)\right), t\right)\right) \]

   14. expm1-define N/A

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \left(\mathsf{expm1}\left(z\right)\right)\right)\right), t\right)\right) \]

   15. expm1-lowering-expm1.f64 97.3%

       \[\leadsto \mathsf{\_.f64}\left(x, \mathsf{/.f64}\left(\mathsf{log1p.f64}\left(\mathsf{*.f64}\left(y, \mathsf{expm1.f64}\left(z\right)\right)\right), t\right)\right) \]

3. Simplified 97.3%

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

5. Taylor expanded in x around inf

   \[\leadsto \color{blue}{x} \]
6. Step-by-step derivation

7. Simplified 71.9%

   \[\leadsto \color{blue}{x} \]
8. Add Preprocessing

Developer Target 1: 75.5% 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 2024158 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -288746230882079470000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- (- x (/ (/ (- 1/2) (* y t)) (* z z))) (* (/ (- 1/2) (* y t)) (/ (/ 2 z) (* z z)))) (- x (/ (log (+ 1 (* z y))) t))))

  (- x (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))