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

Percentage Accurate: 61.1% → 98.3%

Time: 19.3s

Alternatives: 7

Speedup: 211.0×

• could not determine a ground truth

  1. x = 2544628.213875718
  2. y = 1.700894212281153e+277
  3. z = 5.159936187549433e+54
  4. t = -2.896788570852723e+99

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: 61.1% 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 65.5%

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

   1. sub-neg 65.5%

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

   2. associate-+l+ 77.0%

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

   3. cancel-sign-sub 77.0%

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

   4. log1p-define 81.0%

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

   5. cancel-sign-sub 81.0%

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

   6. +-commutative 81.0%

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

   7. unsub-neg 81.0%

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

   8. *-rgt-identity 81.0%

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

   9. distribute-lft-out-- 81.0%

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

   10. expm1-define 98.9%

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

3. Simplified 98.9%

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

5. Final simplification 98.9%

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

Alternative 2: 84.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \mathbf{if}\;y \leq -1.02 \cdot 10^{+251}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+165}:\\ \;\;\;\;x + \frac{-1}{t \cdot 0.5 + \frac{t}{y \cdot \left(e^{z} + -1\right)}}\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+114} \lor \neg \left(y \leq 2.05 \cdot 10^{+142}\right):\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\mathsf{expm1}\left(z\right) \cdot \frac{-1}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (log1p (* y (expm1 z))) (- t))))
   (if (<= y -1.02e+251)
     t_1
     (if (<= y -7.2e+165)
       (+ x (/ -1.0 (+ (* t 0.5) (/ t (* y (+ (exp z) -1.0))))))
       (if (or (<= y -8e+114) (not (<= y 2.05e+142)))
         t_1
         (+ x (* y (* (expm1 z) (/ -1.0 t)))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = log1p((y * expm1(z))) / -t;
	double tmp;
	if (y <= -1.02e+251) {
		tmp = t_1;
	} else if (y <= -7.2e+165) {
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * (exp(z) + -1.0)))));
	} else if ((y <= -8e+114) || !(y <= 2.05e+142)) {
		tmp = t_1;
	} else {
		tmp = x + (y * (expm1(z) * (-1.0 / t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = Math.log1p((y * Math.expm1(z))) / -t;
	double tmp;
	if (y <= -1.02e+251) {
		tmp = t_1;
	} else if (y <= -7.2e+165) {
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * (Math.exp(z) + -1.0)))));
	} else if ((y <= -8e+114) || !(y <= 2.05e+142)) {
		tmp = t_1;
	} else {
		tmp = x + (y * (Math.expm1(z) * (-1.0 / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = math.log1p((y * math.expm1(z))) / -t
	tmp = 0
	if y <= -1.02e+251:
		tmp = t_1
	elif y <= -7.2e+165:
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * (math.exp(z) + -1.0)))))
	elif (y <= -8e+114) or not (y <= 2.05e+142):
		tmp = t_1
	else:
		tmp = x + (y * (math.expm1(z) * (-1.0 / t)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(log1p(Float64(y * expm1(z))) / Float64(-t))
	tmp = 0.0
	if (y <= -1.02e+251)
		tmp = t_1;
	elseif (y <= -7.2e+165)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(t * 0.5) + Float64(t / Float64(y * Float64(exp(z) + -1.0))))));
	elseif ((y <= -8e+114) || !(y <= 2.05e+142))
		tmp = t_1;
	else
		tmp = Float64(x + Float64(y * Float64(expm1(z) * Float64(-1.0 / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t)), $MachinePrecision]}, If[LessEqual[y, -1.02e+251], t$95$1, If[LessEqual[y, -7.2e+165], N[(x + N[(-1.0 / N[(N[(t * 0.5), $MachinePrecision] + N[(t / N[(y * N[(N[Exp[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Or[LessEqual[y, -8e+114], N[Not[LessEqual[y, 2.05e+142]], $MachinePrecision]], t$95$1, N[(x + N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.02e251 or -7.1999999999999996e165 < y < -8e114 or 2.04999999999999991e142 < y

   1. Initial program 30.1%

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

      1. sub-neg 30.1%

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

      2. associate-+l+ 46.0%

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

      3. cancel-sign-sub 46.0%

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

      4. log1p-define 46.0%

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

      5. cancel-sign-sub 46.0%

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

      6. +-commutative 46.0%

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

      7. unsub-neg 46.0%

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

      8. *-rgt-identity 46.0%

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

      9. distribute-lft-out-- 46.0%

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

      10. expm1-define 99.8%

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

   3. Simplified 99.8%

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

   5. Taylor expanded in x around 0 26.8%

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

      1. mul-1-neg 26.8%

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

      2. expm1-define 59.1%

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

      3. log1p-undefine 80.5%

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

      4. distribute-frac-neg2 80.5%

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

   7. Simplified 80.5%

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

   if -1.02e251 < y < -7.1999999999999996e165

   1. Initial program 51.8%

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

      1. sub-neg 51.8%

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

      2. associate-+l+ 92.7%

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

      3. cancel-sign-sub 92.7%

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

      4. log1p-define 92.7%

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

      5. cancel-sign-sub 92.7%

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

      6. +-commutative 92.7%

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

      7. unsub-neg 92.7%

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

      8. *-rgt-identity 92.7%

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

      9. distribute-lft-out-- 92.7%

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

      10. expm1-define 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. Step-by-step derivation

      1. clear-num 99.9%

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

      2. inv-pow 99.9%

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

   6. Applied egg-rr 99.9%

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

      1. unpow-1 99.9%

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

   8. Applied egg-rr 99.9%

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

   9. Taylor expanded in y around 0 81.4%

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

   if -8e114 < y < 2.04999999999999991e142

   1. Initial program 73.7%

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

      1. sub-neg 73.7%

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

      2. associate-+l+ 82.2%

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

      3. cancel-sign-sub 82.2%

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

      4. log1p-define 87.3%

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

      5. cancel-sign-sub 87.3%

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

      6. +-commutative 87.3%

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

      7. unsub-neg 87.3%

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

      8. *-rgt-identity 87.3%

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

      9. distribute-lft-out-- 87.3%

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

      10. expm1-define 98.6%

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

   3. Simplified 98.6%

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

   5. Taylor expanded in y around 0 84.4%

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

      1. expm1-define 95.2%

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

   7. Simplified 95.2%

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

      1. frac-2neg 95.2%

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

      2. div-inv 95.1%

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

      3. distribute-rgt-neg-in 95.1%

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

      4. distribute-neg-frac2 95.1%

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

      5. distribute-neg-frac 95.1%

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

      6. metadata-eval 95.1%

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

   9. Applied egg-rr 95.1%

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

       1. associate-*l* 96.4%

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

   11. Simplified 96.4%

       \[\leadsto x - \color{blue}{y \cdot \left(\left(-\mathsf{expm1}\left(z\right)\right) \cdot \frac{-1}{t}\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 93.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.02 \cdot 10^{+251}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \mathbf{elif}\;y \leq -7.2 \cdot 10^{+165}:\\ \;\;\;\;x + \frac{-1}{t \cdot 0.5 + \frac{t}{y \cdot \left(e^{z} + -1\right)}}\\ \mathbf{elif}\;y \leq -8 \cdot 10^{+114} \lor \neg \left(y \leq 2.05 \cdot 10^{+142}\right):\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\mathsf{expm1}\left(z\right) \cdot \frac{-1}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.999999999995:\\ \;\;\;\;x + \frac{-1}{t \cdot 0.5 + \frac{t}{y \cdot \left(e^{z} + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\mathsf{expm1}\left(z\right) \cdot \frac{-1}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 0.999999999995)
   (+ x (/ -1.0 (+ (* t 0.5) (/ t (* y (+ (exp z) -1.0))))))
   (+ x (* y (* (expm1 z) (/ -1.0 t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 0.999999999995) {
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * (exp(z) + -1.0)))));
	} else {
		tmp = x + (y * (expm1(z) * (-1.0 / t)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.exp(z) <= 0.999999999995) {
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * (Math.exp(z) + -1.0)))));
	} else {
		tmp = x + (y * (Math.expm1(z) * (-1.0 / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.exp(z) <= 0.999999999995:
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * (math.exp(z) + -1.0)))))
	else:
		tmp = x + (y * (math.expm1(z) * (-1.0 / t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (exp(z) <= 0.999999999995)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(t * 0.5) + Float64(t / Float64(y * Float64(exp(z) + -1.0))))));
	else
		tmp = Float64(x + Float64(y * Float64(expm1(z) * Float64(-1.0 / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.999999999995], N[(x + N[(-1.0 / N[(N[(t * 0.5), $MachinePrecision] + N[(t / N[(y * N[(N[Exp[z], $MachinePrecision] + -1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.999999999995

   1. Initial program 87.2%

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

      1. sub-neg 87.2%

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

      2. associate-+l+ 87.2%

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

      3. cancel-sign-sub 87.2%

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

      4. log1p-define 99.3%

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

      5. cancel-sign-sub 99.3%

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

      6. +-commutative 99.3%

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

      7. unsub-neg 99.3%

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

      8. *-rgt-identity 99.3%

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

      9. distribute-lft-out-- 99.3%

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

      10. expm1-define 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.8%

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

      2. inv-pow 99.8%

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

   6. Applied egg-rr 99.8%

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

      1. unpow-1 99.8%

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

   8. Applied egg-rr 99.8%

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

   9. Taylor expanded in y around 0 79.8%

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

   if 0.999999999995 < (exp.f64 z)

   1. Initial program 55.7%

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

      1. sub-neg 55.7%

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

      2. associate-+l+ 72.3%

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

      3. cancel-sign-sub 72.3%

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

      4. log1p-define 72.7%

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

      5. cancel-sign-sub 72.7%

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

      6. +-commutative 72.7%

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

      7. unsub-neg 72.7%

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

      8. *-rgt-identity 72.7%

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

      9. distribute-lft-out-- 72.7%

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

      10. expm1-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in y around 0 72.7%

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

      1. expm1-define 88.8%

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

   7. Simplified 88.8%

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

      1. frac-2neg 88.8%

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

      2. div-inv 88.7%

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

      3. distribute-rgt-neg-in 88.7%

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

      4. distribute-neg-frac2 88.7%

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

      5. distribute-neg-frac 88.7%

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

      6. metadata-eval 88.7%

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

   9. Applied egg-rr 88.7%

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

       1. associate-*l* 90.3%

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

   11. Simplified 90.3%

       \[\leadsto x - \color{blue}{y \cdot \left(\left(-\mathsf{expm1}\left(z\right)\right) \cdot \frac{-1}{t}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 87.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0.999999999995:\\ \;\;\;\;x + \frac{-1}{t \cdot 0.5 + \frac{t}{y \cdot \left(e^{z} + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(\mathsf{expm1}\left(z\right) \cdot \frac{-1}{t}\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 85.5% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. sub-neg 65.5%

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

   2. associate-+l+ 77.0%

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

   3. cancel-sign-sub 77.0%

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

   4. log1p-define 81.0%

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

   5. cancel-sign-sub 81.0%

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

   6. +-commutative 81.0%

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

   7. unsub-neg 81.0%

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

   8. *-rgt-identity 81.0%

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

   9. distribute-lft-out-- 81.0%

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

   10. expm1-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in y around 0 73.1%

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

   1. expm1-define 84.2%

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

7. Simplified 84.2%

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

   1. frac-2neg 84.2%

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

   2. div-inv 84.1%

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

   3. distribute-rgt-neg-in 84.1%

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

   4. distribute-neg-frac2 84.1%

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

   5. distribute-neg-frac 84.1%

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

   6. metadata-eval 84.1%

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

9. Applied egg-rr 84.1%

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

    1. associate-*l* 85.2%

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

11. Simplified 85.2%

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

12. Final simplification 85.2%

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

Alternative 5: 84.5% accurate, 2.0× speedup

Math

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

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(Float64(y * expm1(z)) / t))
end

Wolfram

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

Derivation

1. Initial program 65.5%

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

   1. sub-neg 65.5%

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

   2. associate-+l+ 77.0%

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

   3. cancel-sign-sub 77.0%

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

   4. log1p-define 81.0%

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

   5. cancel-sign-sub 81.0%

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

   6. +-commutative 81.0%

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

   7. unsub-neg 81.0%

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

   8. *-rgt-identity 81.0%

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

   9. distribute-lft-out-- 81.0%

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

   10. expm1-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in y around 0 73.1%

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

   1. expm1-define 84.2%

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

7. Simplified 84.2%

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

8. Final simplification 84.2%

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

Alternative 6: 82.1% accurate, 17.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.4e+27) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.4e+27) {
		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 <= (-5.4d+27)) 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 <= -5.4e+27) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5.4e+27:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.4e+27)
		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 <= -5.4e+27)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.4e+27], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.3999999999999995e27

   1. Initial program 88.1%

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

      1. sub-neg 88.1%

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

      2. associate-+l+ 88.1%

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

      3. cancel-sign-sub 88.1%

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

      4. log1p-define 99.9%

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

      5. cancel-sign-sub 99.9%

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

      6. +-commutative 99.9%

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

      7. unsub-neg 99.9%

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

      8. *-rgt-identity 99.9%

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

      9. distribute-lft-out-- 99.9%

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

      10. expm1-define 99.9%

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

   3. Simplified 99.9%

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

   5. Taylor expanded in x around inf 66.4%

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

   if -5.3999999999999995e27 < z

   1. Initial program 56.3%

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

      1. sub-neg 56.3%

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

      2. associate-+l+ 72.4%

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

      3. cancel-sign-sub 72.4%

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

      4. log1p-define 73.3%

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

      5. cancel-sign-sub 73.3%

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

      6. +-commutative 73.3%

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

      7. unsub-neg 73.3%

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

      8. *-rgt-identity 73.3%

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

      9. distribute-lft-out-- 73.3%

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

      10. expm1-define 98.5%

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

   3. Simplified 98.5%

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

   5. Taylor expanded in z around 0 87.4%

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

      1. associate-/l* 88.9%

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

   7. Simplified 88.9%

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

4. Final simplification 82.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.4 \cdot 10^{+27}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.6% 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 65.5%

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

   1. sub-neg 65.5%

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

   2. associate-+l+ 77.0%

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

   3. cancel-sign-sub 77.0%

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

   4. log1p-define 81.0%

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

   5. cancel-sign-sub 81.0%

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

   6. +-commutative 81.0%

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

   7. unsub-neg 81.0%

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

   8. *-rgt-identity 81.0%

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

   9. distribute-lft-out-- 81.0%

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

   10. expm1-define 98.9%

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

3. Simplified 98.9%

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

5. Taylor expanded in x around inf 70.1%

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

6. Final simplification 70.1%

   \[\leadsto x \]
7. Add Preprocessing

Developer target: 75.0% 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 2024039 
(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)))