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

Percentage Accurate: 61.8% → 98.5%

Time: 15.1s

Alternatives: 9

Speedup: 211.0×

• could not determine a ground truth

  1. x = -2.0239062369290676e-184
  2. y = 5.272267179618203e+101
  3. z = 2.0834499819507277e+45
  4. t = -1.9315047314778553e-271

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: 98.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{t} \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (- x (/ (log1p (* y (expm1 z))) t)))

C

double code(double x, double y, double z, double t) {
	return x - (log1p((y * expm1(z))) / t);
}

Java

public static double code(double x, double y, double z, double t) {
	return x - (Math.log1p((y * Math.expm1(z))) / t);
}

Python

def code(x, y, z, t):
	return x - (math.log1p((y * math.expm1(z))) / t)

Julia

function code(x, y, z, t)
	return Float64(x - Float64(log1p(Float64(y * expm1(z))) / t))
end

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. sub-neg 61.7%

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

   2. associate-+l+ 79.9%

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

   3. cancel-sign-sub 79.9%

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

   4. log1p-def 85.3%

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

   5. cancel-sign-sub 85.3%

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

   6. +-commutative 85.3%

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

   7. unsub-neg 85.3%

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

   8. *-rgt-identity 85.3%

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

   9. distribute-lft-out-- 85.3%

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

   10. expm1-def 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. Final simplification 98.6%

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

Alternative 2: 90.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{t \cdot 0.5 + \frac{t}{y \cdot \left(e^{z} + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{t \cdot 0.5 + \frac{\frac{t}{z} - t \cdot 0.5}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (exp z) 0.0)
   (+ x (/ -1.0 (+ (* t 0.5) (/ t (* y (+ (exp z) -1.0))))))
   (+ x (/ -1.0 (+ (* t 0.5) (/ (- (/ t z) (* t 0.5)) y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (exp(z) <= 0.0) {
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * (exp(z) + -1.0)))));
	} else {
		tmp = x + (-1.0 / ((t * 0.5) + (((t / z) - (t * 0.5)) / y)));
	}
	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 (exp(z) <= 0.0d0) then
        tmp = x + ((-1.0d0) / ((t * 0.5d0) + (t / (y * (exp(z) + (-1.0d0))))))
    else
        tmp = x + ((-1.0d0) / ((t * 0.5d0) + (((t / z) - (t * 0.5d0)) / y)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (Math.exp(z) <= 0.0) {
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * (Math.exp(z) + -1.0)))));
	} else {
		tmp = x + (-1.0 / ((t * 0.5) + (((t / z) - (t * 0.5)) / y)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if math.exp(z) <= 0.0:
		tmp = x + (-1.0 / ((t * 0.5) + (t / (y * (math.exp(z) + -1.0)))))
	else:
		tmp = x + (-1.0 / ((t * 0.5) + (((t / z) - (t * 0.5)) / y)))
	return tmp

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[Exp[z], $MachinePrecision], 0.0], 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[(-1.0 / N[(N[(t * 0.5), $MachinePrecision] + N[(N[(N[(t / z), $MachinePrecision] - N[(t * 0.5), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (exp.f64 z) < 0.0

   1. Initial program 83.4%

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

      1. sub-neg 83.4%

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

      2. associate-+l+ 83.4%

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

      3. cancel-sign-sub 83.4%

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

      4. log1p-def 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-def 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.9%

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

      2. associate-/r/ 99.9%

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

   5. Applied egg-rr 99.9%

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

      1. associate-/r/ 99.9%

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

   7. Applied egg-rr 99.9%

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

   8. Taylor expanded in y around 0 90.9%

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

   if 0.0 < (exp.f64 z)

   1. Initial program 52.9%

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

      1. sub-neg 52.9%

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

      2. associate-+l+ 78.5%

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

      3. cancel-sign-sub 78.5%

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

      4. log1p-def 79.4%

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

      5. cancel-sign-sub 79.4%

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

      6. +-commutative 79.4%

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

      7. unsub-neg 79.4%

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

      8. *-rgt-identity 79.4%

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

      9. distribute-lft-out-- 79.4%

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

      10. expm1-def 98.0%

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

   3. Simplified 98.0%

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

      1. clear-num 97.9%

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

      2. associate-/r/ 98.0%

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

   5. Applied egg-rr 98.0%

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

      1. associate-/r/ 97.9%

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

   7. Applied egg-rr 97.9%

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

   8. Taylor expanded in z around 0 59.0%

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

      1. fma-def 59.0%

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

      2. associate-/l* 65.2%

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

      3. unpow2 65.2%

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

      4. mul-1-neg 65.2%

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

      5. unpow2 65.2%

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

   10. Simplified 65.2%

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

   11. Taylor expanded in y around -inf 90.8%

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

       1. +-commutative 90.8%

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

       2. mul-1-neg 90.8%

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

       3. unsub-neg 90.8%

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

       4. *-commutative 90.8%

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

       5. +-commutative 90.8%

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

       6. mul-1-neg 90.8%

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

       7. unsub-neg 90.8%

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

       8. *-commutative 90.8%

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

   13. Simplified 90.8%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \leq 0:\\ \;\;\;\;x + \frac{-1}{t \cdot 0.5 + \frac{t}{y \cdot \left(e^{z} + -1\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{t \cdot 0.5 + \frac{\frac{t}{z} - t \cdot 0.5}{y}}\\ \end{array} \]

Alternative 3: 89.7% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Python

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

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -0.5)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(-0.5 * Float64(t / y)) + Float64(Float64(t * 0.5) + Float64(t / Float64(y * z))))));
	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[y, -0.5], N[(x + N[(-1.0 / N[(N[(-0.5 * N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(N[(t * 0.5), $MachinePrecision] + N[(t / N[(y * z), $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 y < -0.5

   1. Initial program 37.7%

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

      1. sub-neg 37.7%

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

      2. associate-+l+ 77.1%

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

      3. cancel-sign-sub 77.1%

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

      4. log1p-def 77.1%

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

      5. cancel-sign-sub 77.1%

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

      6. +-commutative 77.1%

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

      7. unsub-neg 77.1%

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

      8. *-rgt-identity 77.1%

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

      9. distribute-lft-out-- 77.1%

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

      10. expm1-def 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around 0 35.7%

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

      1. fma-def 35.7%

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

      2. associate-/l* 37.2%

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

      3. unpow2 37.2%

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

      4. mul-1-neg 37.2%

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

      5. unpow2 37.2%

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

   10. Simplified 37.2%

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

   11. Taylor expanded in y around 0 78.4%

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

   if -0.5 < y

   1. Initial program 70.1%

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

      1. sub-neg 70.1%

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

      2. associate-+l+ 80.9%

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

      3. cancel-sign-sub 80.9%

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

      4. log1p-def 88.2%

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

      5. cancel-sign-sub 88.2%

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

      6. +-commutative 88.2%

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

      7. unsub-neg 88.2%

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

      8. *-rgt-identity 88.2%

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

      9. distribute-lft-out-- 88.2%

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

      10. expm1-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in y around 0 86.7%

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

      1. expm1-def 93.4%

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

   6. Simplified 93.4%

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

      1. div-inv 93.4%

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

      2. associate-*l* 94.7%

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

   8. Applied egg-rr 94.7%

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

4. Final simplification 90.5%

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

Alternative 4: 89.8% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -0.98:\\ \;\;\;\;x + \frac{-1}{-0.5 \cdot \frac{t}{y} + \left(t \cdot 0.5 + \frac{t}{y \cdot z}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -0.98)
   (+ x (/ -1.0 (+ (* -0.5 (/ t y)) (+ (* t 0.5) (/ t (* y z))))))
   (- x (/ y (/ t (expm1 z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -0.98) {
		tmp = x + (-1.0 / ((-0.5 * (t / y)) + ((t * 0.5) + (t / (y * z)))));
	} else {
		tmp = x - (y / (t / expm1(z)));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -0.98) {
		tmp = x + (-1.0 / ((-0.5 * (t / y)) + ((t * 0.5) + (t / (y * z)))));
	} else {
		tmp = x - (y / (t / Math.expm1(z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -0.98:
		tmp = x + (-1.0 / ((-0.5 * (t / y)) + ((t * 0.5) + (t / (y * z)))))
	else:
		tmp = x - (y / (t / math.expm1(z)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -0.98)
		tmp = Float64(x + Float64(-1.0 / Float64(Float64(-0.5 * Float64(t / y)) + Float64(Float64(t * 0.5) + Float64(t / Float64(y * z))))));
	else
		tmp = Float64(x - Float64(y / Float64(t / expm1(z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -0.98], N[(x + N[(-1.0 / N[(N[(-0.5 * N[(t / y), $MachinePrecision]), $MachinePrecision] + N[(N[(t * 0.5), $MachinePrecision] + N[(t / N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -0.97999999999999998

   1. Initial program 37.7%

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

      1. sub-neg 37.7%

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

      2. associate-+l+ 77.1%

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

      3. cancel-sign-sub 77.1%

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

      4. log1p-def 77.1%

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

      5. cancel-sign-sub 77.1%

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

      6. +-commutative 77.1%

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

      7. unsub-neg 77.1%

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

      8. *-rgt-identity 77.1%

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

      9. distribute-lft-out-- 77.1%

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

      10. expm1-def 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around 0 35.7%

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

      1. fma-def 35.7%

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

      2. associate-/l* 37.2%

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

      3. unpow2 37.2%

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

      4. mul-1-neg 37.2%

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

      5. unpow2 37.2%

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

   10. Simplified 37.2%

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

   11. Taylor expanded in y around 0 78.4%

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

   if -0.97999999999999998 < y

   1. Initial program 70.1%

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

      1. sub-neg 70.1%

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

      2. associate-+l+ 80.9%

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

      3. cancel-sign-sub 80.9%

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

      4. log1p-def 88.2%

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

      5. cancel-sign-sub 88.2%

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

      6. +-commutative 88.2%

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

      7. unsub-neg 88.2%

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

      8. *-rgt-identity 88.2%

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

      9. distribute-lft-out-- 88.2%

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

      10. expm1-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in y around 0 86.7%

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

      1. associate-/l* 86.7%

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

      2. expm1-def 94.7%

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

   6. Simplified 94.7%

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

4. Final simplification 90.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -0.98:\\ \;\;\;\;x + \frac{-1}{-0.5 \cdot \frac{t}{y} + \left(t \cdot 0.5 + \frac{t}{y \cdot z}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{\mathsf{expm1}\left(z\right)}}\\ \end{array} \]

Alternative 5: 85.4% accurate, 12.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. sub-neg 61.7%

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

   2. associate-+l+ 79.9%

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

   3. cancel-sign-sub 79.9%

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

   4. log1p-def 85.3%

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

   5. cancel-sign-sub 85.3%

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

   6. +-commutative 85.3%

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

   7. unsub-neg 85.3%

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

   8. *-rgt-identity 85.3%

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

   9. distribute-lft-out-- 85.3%

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

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

   1. clear-num 98.5%

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

   2. associate-/r/ 98.5%

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

5. Applied egg-rr 98.5%

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

   1. associate-/r/ 98.5%

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

7. Applied egg-rr 98.5%

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

8. Taylor expanded in z around 0 56.4%

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

   1. fma-def 56.4%

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

   2. associate-/l* 64.1%

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

   3. unpow2 64.1%

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

   4. mul-1-neg 64.1%

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

   5. unpow2 64.1%

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

10. Simplified 64.1%

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

11. Taylor expanded in y around -inf 86.2%

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

    1. +-commutative 86.2%

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

    2. mul-1-neg 86.2%

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

    3. unsub-neg 86.2%

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

    4. *-commutative 86.2%

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

    5. +-commutative 86.2%

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

    6. mul-1-neg 86.2%

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

    7. unsub-neg 86.2%

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

    8. *-commutative 86.2%

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

13. Simplified 86.2%

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

14. Final simplification 86.2%

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

Alternative 6: 82.6% accurate, 16.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+187}:\\ \;\;\;\;x - \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z} + t \cdot -0.5}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.05e+187) (- x (/ 2.0 t)) (- x (/ y (+ (/ t z) (* t -0.5))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.05e+187) {
		tmp = x - (2.0 / t);
	} else {
		tmp = x - (y / ((t / z) + (t * -0.5)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.05d+187)) then
        tmp = x - (2.0d0 / t)
    else
        tmp = x - (y / ((t / z) + (t * (-0.5d0))))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.05e+187) {
		tmp = x - (2.0 / t);
	} else {
		tmp = x - (y / ((t / z) + (t * -0.5)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.05e+187:
		tmp = x - (2.0 / t)
	else:
		tmp = x - (y / ((t / z) + (t * -0.5)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.05e+187)
		tmp = Float64(x - Float64(2.0 / t));
	else
		tmp = Float64(x - Float64(y / Float64(Float64(t / z) + Float64(t * -0.5))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.05e+187)
		tmp = x - (2.0 / t);
	else
		tmp = x - (y / ((t / z) + (t * -0.5)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.05e+187], N[(x - N[(2.0 / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y / N[(N[(t / z), $MachinePrecision] + N[(t * -0.5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e187

   1. Initial program 41.0%

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

      1. sub-neg 41.0%

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

      2. associate-+l+ 68.0%

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

      3. cancel-sign-sub 68.0%

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

      4. log1p-def 68.0%

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

      5. cancel-sign-sub 68.0%

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

      6. +-commutative 68.0%

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

      7. unsub-neg 68.0%

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

      8. *-rgt-identity 68.0%

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

      9. distribute-lft-out-- 68.0%

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

      10. expm1-def 99.9%

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

   3. Simplified 99.9%

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

      1. clear-num 99.7%

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

      2. associate-/r/ 99.8%

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

   5. Applied egg-rr 99.8%

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

      1. associate-/r/ 99.7%

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

   7. Applied egg-rr 99.7%

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

   8. Taylor expanded in z around 0 0.0%

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

      1. fma-def 0.0%

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

      2. associate-/l* 0.0%

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

      3. unpow2 0.0%

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

      4. mul-1-neg 0.0%

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

      5. unpow2 0.0%

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

   10. Simplified 0.0%

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

   11. Taylor expanded in y around inf 59.6%

       \[\leadsto x - \color{blue}{\frac{2}{t}} \]

   if -1.05e187 < y

   1. Initial program 64.5%

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

      1. sub-neg 64.5%

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

      2. associate-+l+ 81.5%

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

      3. cancel-sign-sub 81.5%

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

      4. log1p-def 87.6%

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

      5. cancel-sign-sub 87.6%

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

      6. +-commutative 87.6%

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

      7. unsub-neg 87.6%

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

      8. *-rgt-identity 87.6%

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

      9. distribute-lft-out-- 87.6%

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

      10. expm1-def 98.4%

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

   3. Simplified 98.4%

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

   4. Taylor expanded in y around 0 84.8%

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

      1. associate-/l* 84.8%

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

      2. expm1-def 93.1%

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

   6. Simplified 93.1%

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

   7. Taylor expanded in z around 0 87.9%

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

4. Final simplification 84.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+187}:\\ \;\;\;\;x - \frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y}{\frac{t}{z} + t \cdot -0.5}\\ \end{array} \]

Alternative 7: 82.5% accurate, 19.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -100000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \frac{-1}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -100000000000.0) x (+ x (* y (* z (/ -1.0 t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -100000000000.0) {
		tmp = x;
	} else {
		tmp = x + (y * (z * (-1.0 / t)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-100000000000.0d0)) then
        tmp = x
    else
        tmp = x + (y * (z * ((-1.0d0) / t)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -100000000000.0) {
		tmp = x;
	} else {
		tmp = x + (y * (z * (-1.0 / t)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -100000000000.0:
		tmp = x
	else:
		tmp = x + (y * (z * (-1.0 / t)))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -100000000000.0)
		tmp = x;
	else
		tmp = Float64(x + Float64(y * Float64(z * Float64(-1.0 / t))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -100000000000.0)
		tmp = x;
	else
		tmp = x + (y * (z * (-1.0 / t)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -100000000000.0], x, N[(x + N[(y * N[(z * N[(-1.0 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1e11

   1. Initial program 82.9%

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

      1. sub-neg 82.9%

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

      2. associate-+l+ 82.9%

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

      3. cancel-sign-sub 82.9%

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

      4. log1p-def 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-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 69.7%

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

   if -1e11 < z

   1. Initial program 53.4%

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

      1. sub-neg 53.4%

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

      2. associate-+l+ 78.7%

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

      3. cancel-sign-sub 78.7%

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

      4. log1p-def 79.6%

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

      5. cancel-sign-sub 79.6%

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

      6. +-commutative 79.6%

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

      7. unsub-neg 79.6%

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

      8. *-rgt-identity 79.6%

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

      9. distribute-lft-out-- 79.6%

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

      10. expm1-def 98.1%

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

   3. Simplified 98.1%

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

   4. Taylor expanded in z around 0 87.6%

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

      1. associate-/l* 88.3%

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

      2. associate-/r/ 86.9%

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

   6. Simplified 86.9%

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

      1. associate-*l/ 87.6%

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

      2. clear-num 87.5%

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

   8. Applied egg-rr 87.5%

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

      1. associate-/r/ 87.6%

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

      2. *-commutative 87.6%

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

      3. associate-*r* 88.3%

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

   10. Applied egg-rr 88.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -100000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x + y \cdot \left(z \cdot \frac{-1}{t}\right)\\ \end{array} \]

Alternative 8: 82.5% accurate, 23.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -66000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -66000000000.0) x (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -66000000000.0) {
		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 <= (-66000000000.0d0)) 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 <= -66000000000.0) {
		tmp = x;
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -66000000000.0:
		tmp = x
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -66000000000.0)
		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 <= -66000000000.0)
		tmp = x;
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -66000000000.0], x, N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.6e10

   1. Initial program 82.9%

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

      1. sub-neg 82.9%

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

      2. associate-+l+ 82.9%

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

      3. cancel-sign-sub 82.9%

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

      4. log1p-def 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-def 99.9%

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

   3. Simplified 99.9%

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

   4. Taylor expanded in x around inf 69.7%

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

   if -6.6e10 < z

   1. Initial program 53.4%

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

      1. sub-neg 53.4%

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

      2. associate-+l+ 78.7%

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

      3. cancel-sign-sub 78.7%

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

      4. log1p-def 79.6%

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

      5. cancel-sign-sub 79.6%

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

      6. +-commutative 79.6%

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

      7. unsub-neg 79.6%

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

      8. *-rgt-identity 79.6%

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

      9. distribute-lft-out-- 79.6%

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

      10. expm1-def 98.1%

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

   3. Simplified 98.1%

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

      1. *-un-lft-identity 98.1%

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

      2. add-cube-cbrt 97.6%

         \[\leadsto x - \frac{1 \cdot \mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{\color{blue}{\left(\sqrt[3]{t} \cdot \sqrt[3]{t}\right) \cdot \sqrt[3]{t}}} \]

      3. times-frac 97.6%

         \[\leadsto x - \color{blue}{\frac{1}{\sqrt[3]{t} \cdot \sqrt[3]{t}} \cdot \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{\sqrt[3]{t}}} \]

      4. pow2 97.6%

         \[\leadsto x - \frac{1}{\color{blue}{{\left(\sqrt[3]{t}\right)}^{2}}} \cdot \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{\sqrt[3]{t}} \]

   5. Applied egg-rr 97.6%

      \[\leadsto x - \color{blue}{\frac{1}{{\left(\sqrt[3]{t}\right)}^{2}} \cdot \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{\sqrt[3]{t}}} \]
   6. Step-by-step derivation

      1. associate-*l/ 97.6%

         \[\leadsto x - \color{blue}{\frac{1 \cdot \frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{\sqrt[3]{t}}}{{\left(\sqrt[3]{t}\right)}^{2}}} \]

      2. *-lft-identity 97.6%

         \[\leadsto x - \frac{\color{blue}{\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{\sqrt[3]{t}}}}{{\left(\sqrt[3]{t}\right)}^{2}} \]

   7. Simplified 97.6%

      \[\leadsto x - \color{blue}{\frac{\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{\sqrt[3]{t}}}{{\left(\sqrt[3]{t}\right)}^{2}}} \]

   8. Taylor expanded in z around 0 87.6%

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

      1. associate-*r/ 88.3%

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

   10. Simplified 88.3%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -66000000000:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \]

Alternative 9: 72.1% accurate, 211.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 61.7%

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

   1. sub-neg 61.7%

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

   2. associate-+l+ 79.9%

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

   3. cancel-sign-sub 79.9%

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

   4. log1p-def 85.3%

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

   5. cancel-sign-sub 85.3%

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

   6. +-commutative 85.3%

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

   7. unsub-neg 85.3%

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

   8. *-rgt-identity 85.3%

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

   9. distribute-lft-out-- 85.3%

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

   10. expm1-def 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. Taylor expanded in x around inf 75.4%

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

5. Final simplification 75.4%

   \[\leadsto x \]

Developer target: 74.9% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2023293 
(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)))