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

Percentage Accurate: 61.9% → 98.6%

Time: 12.6s

Alternatives: 7

Speedup: 30.1×

• could not determine a ground truth

  1. x = -8.556658036576619e+281
  2. y = 4.038928316643811e-107
  3. z = 1.340350827545086e+127
  4. t = -6.938147416049015e-96

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: 61.9% 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: 98.6% 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 62.8%

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

   1. associate-+l- 79.0%

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

   2. sub-neg 79.0%

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

   3. log1p-def 84.7%

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

   4. neg-sub0 84.7%

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

   5. associate-+l- 84.7%

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

   6. neg-sub0 84.7%

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

   7. neg-mul-1 84.7%

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

   8. *-commutative 84.7%

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

   9. distribute-rgt-out 84.7%

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

   10. +-commutative 84.7%

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

   11. metadata-eval 84.7%

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

   12. sub-neg 84.7%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

4. Final simplification 99.0%

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

Alternative 2: 88.8% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. associate-+l- 79.0%

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

   2. sub-neg 79.0%

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

   3. log1p-def 84.7%

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

   4. neg-sub0 84.7%

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

   5. associate-+l- 84.7%

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

   6. neg-sub0 84.7%

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

   7. neg-mul-1 84.7%

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

   8. *-commutative 84.7%

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

   9. distribute-rgt-out 84.7%

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

   10. +-commutative 84.7%

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

   11. metadata-eval 84.7%

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

   12. sub-neg 84.7%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 98.9%

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

   2. associate-/r/ 98.9%

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

5. Applied egg-rr 98.9%

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

   1. associate-/r/ 98.9%

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

7. Applied egg-rr 98.9%

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

   1. div-inv 98.6%

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

9. Applied egg-rr 98.6%

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

10. Taylor expanded in y around 0 79.0%

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

    1. associate-/l/ 79.0%

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

    2. expm1-def 89.2%

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

12. Simplified 89.2%

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

13. Final simplification 89.2%

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

Alternative 3: 89.0% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. associate-+l- 79.0%

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

   2. sub-neg 79.0%

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

   3. log1p-def 84.7%

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

   4. neg-sub0 84.7%

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

   5. associate-+l- 84.7%

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

   6. neg-sub0 84.7%

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

   7. neg-mul-1 84.7%

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

   8. *-commutative 84.7%

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

   9. distribute-rgt-out 84.7%

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

   10. +-commutative 84.7%

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

   11. metadata-eval 84.7%

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

   12. sub-neg 84.7%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

   1. clear-num 98.9%

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

   2. associate-/r/ 98.9%

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

5. Applied egg-rr 98.9%

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

   1. associate-/r/ 98.9%

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

7. Applied egg-rr 98.9%

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

8. Taylor expanded in y around 0 79.0%

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

   1. expm1-def 89.5%

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

   2. *-commutative 89.5%

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

10. Simplified 89.5%

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

11. Final simplification 89.5%

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

Alternative 4: 85.6% accurate, 2.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. associate-+l- 79.0%

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

   2. sub-neg 79.0%

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

   3. log1p-def 84.7%

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

   4. neg-sub0 84.7%

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

   5. associate-+l- 84.7%

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

   6. neg-sub0 84.7%

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

   7. neg-mul-1 84.7%

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

   8. *-commutative 84.7%

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

   9. distribute-rgt-out 84.7%

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

   10. +-commutative 84.7%

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

   11. metadata-eval 84.7%

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

   12. sub-neg 84.7%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in y around 0 78.1%

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

   1. associate-/l* 77.6%

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

   2. associate-/r/ 78.1%

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

   3. expm1-def 87.8%

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

6. Simplified 87.8%

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

7. Final simplification 87.8%

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

Alternative 5: 72.7% accurate, 30.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - (z * (y / 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 - (z * (y / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. associate-+l- 79.0%

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

   2. sub-neg 79.0%

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

   3. log1p-def 84.7%

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

   4. neg-sub0 84.7%

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

   5. associate-+l- 84.7%

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

   6. neg-sub0 84.7%

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

   7. neg-mul-1 84.7%

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

   8. *-commutative 84.7%

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

   9. distribute-rgt-out 84.7%

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

   10. +-commutative 84.7%

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

   11. metadata-eval 84.7%

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

   12. sub-neg 84.7%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 76.8%

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

   1. associate-/l* 76.9%

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

   2. associate-/r/ 75.2%

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

6. Simplified 75.2%

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

7. Final simplification 75.2%

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

Alternative 6: 74.9% accurate, 30.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x - (y * (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 - (y * (z / t))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. associate-+l- 79.0%

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

   2. sub-neg 79.0%

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

   3. log1p-def 84.7%

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

   4. neg-sub0 84.7%

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

   5. associate-+l- 84.7%

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

   6. neg-sub0 84.7%

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

   7. neg-mul-1 84.7%

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

   8. *-commutative 84.7%

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

   9. distribute-rgt-out 84.7%

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

   10. +-commutative 84.7%

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

   11. metadata-eval 84.7%

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

   12. sub-neg 84.7%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 76.8%

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

   1. associate-/l* 76.9%

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

   2. associate-/r/ 75.2%

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

6. Simplified 75.2%

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

   1. expm1-log1p-u 71.1%

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

   2. expm1-udef 68.8%

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

   3. *-commutative 68.8%

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

   4. clear-num 68.8%

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

   5. un-div-inv 68.8%

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

8. Applied egg-rr 68.8%

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

   1. expm1-def 71.4%

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

   2. expm1-log1p 76.0%

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

   3. associate-/r/ 76.8%

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

10. Simplified 76.8%

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

11. Final simplification 76.8%

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

Alternative 7: 75.0% accurate, 30.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.8%

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

   1. associate-+l- 79.0%

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

   2. sub-neg 79.0%

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

   3. log1p-def 84.7%

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

   4. neg-sub0 84.7%

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

   5. associate-+l- 84.7%

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

   6. neg-sub0 84.7%

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

   7. neg-mul-1 84.7%

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

   8. *-commutative 84.7%

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

   9. distribute-rgt-out 84.7%

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

   10. +-commutative 84.7%

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

   11. metadata-eval 84.7%

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

   12. sub-neg 84.7%

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

   13. expm1-def 99.0%

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

3. Simplified 99.0%

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

4. Taylor expanded in z around 0 76.8%

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

   1. associate-/l* 76.9%

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

   2. associate-/r/ 75.2%

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

6. Simplified 75.2%

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

   1. associate-*l/ 76.8%

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

   2. associate-/l* 76.9%

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

8. Applied egg-rr 76.9%

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

9. Final simplification 76.9%

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

Developer target: 74.7% 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 2023257 
(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)))