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

Percentage Accurate: 61.0% → 98.5%

Time: 20.2s

Alternatives: 10

Speedup: 11.3×

• could not determine a ground truth

  1. x = -8.111510039454363e-148
  2. y = 2.9316332681477266e+237
  3. z = 6.80503116902961e+171
  4. t = -1.0430007988175546e-173

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: 61.0% 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 65.1%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. distribute-lft-out-- N/A

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

   6. lower-log1p.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-expm1.f64 98.8

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

5. Applied rewrites 98.8%

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

Alternative 2: 95.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ t_2 := \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+282}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-77}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(y, t \cdot 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{+274}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- x (/ (log1p (* y z)) t)))
        (t_2 (/ (log (+ (- 1.0 y) (* y (exp z)))) t)))
   (if (<= t_2 -5e+282)
     t_1
     (if (<= t_2 2e-77)
       (+ x (/ -1.0 (/ (fma y (* t 0.5) (/ t (expm1 z))) y)))
       (if (<= t_2 4e+274) (/ (log1p (* y (expm1 z))) (- t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x - (log1p((y * z)) / t);
	double t_2 = log(((1.0 - y) + (y * exp(z)))) / t;
	double tmp;
	if (t_2 <= -5e+282) {
		tmp = t_1;
	} else if (t_2 <= 2e-77) {
		tmp = x + (-1.0 / (fma(y, (t * 0.5), (t / expm1(z))) / y));
	} else if (t_2 <= 4e+274) {
		tmp = log1p((y * expm1(z))) / -t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(x - Float64(log1p(Float64(y * z)) / t))
	t_2 = Float64(log(Float64(Float64(1.0 - y) + Float64(y * exp(z)))) / t)
	tmp = 0.0
	if (t_2 <= -5e+282)
		tmp = t_1;
	elseif (t_2 <= 2e-77)
		tmp = Float64(x + Float64(-1.0 / Float64(fma(y, Float64(t * 0.5), Float64(t / expm1(z))) / y)));
	elseif (t_2 <= 4e+274)
		tmp = Float64(log1p(Float64(y * expm1(z))) / Float64(-t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[Log[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+282], t$95$1, If[LessEqual[t$95$2, 2e-77], N[(x + N[(-1.0 / N[(N[(y * N[(t * 0.5), $MachinePrecision] + N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e+274], N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t)), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) t) < -4.99999999999999978e282 or 3.99999999999999969e274 < (/.f64 (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) t)

   1. Initial program 4.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. distribute-lft-out-- N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 98.7%

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

   if -4.99999999999999978e282 < (/.f64 (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) t) < 1.9999999999999999e-77

   1. Initial program 87.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 75.8

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

   5. Applied rewrites 75.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 75.8

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

   7. Applied rewrites 75.8%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-expm1.f64 96.3

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

   10. Applied rewrites 96.3%

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

   if 1.9999999999999999e-77 < (/.f64 (log.f64 (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))) t) < 3.99999999999999969e274

   1. Initial program 98.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. *-rgt-identity N/A

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

      7. cancel-sign-sub-inv N/A

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

      8. distribute-lft-out-- N/A

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

      9. lower-log1p.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-neg.f64 82.5

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

   5. Applied rewrites 82.5%

      \[\leadsto \color{blue}{\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \leq -5 \cdot 10^{+282}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \leq 2 \cdot 10^{-77}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(y, t \cdot 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \mathbf{elif}\;\frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \leq 4 \cdot 10^{+274}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 94.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(y, t \cdot 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- 1.0 y) (* y (exp z))) 0.0)
   (- x (/ (log1p (* y z)) t))
   (+ x (/ -1.0 (/ (fma y (* t 0.5) (/ t (expm1 z))) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((1.0 - y) + (y * exp(z))) <= 0.0) {
		tmp = x - (log1p((y * z)) / t);
	} else {
		tmp = x + (-1.0 / (fma(y, (t * 0.5), (t / expm1(z))) / y));
	}
	return tmp;
}

Julia

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 / N[(N[(y * N[(t * 0.5), $MachinePrecision] + N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 0.0

   1. Initial program 1.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. distribute-lft-out-- N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 99.9%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 88.0%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 73.1

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

   5. Applied rewrites 73.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 73.1

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

   7. Applied rewrites 73.1%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

      6. distribute-rgt-neg-in N/A

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

      7. *-commutative N/A

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

      8. distribute-lft-neg-in N/A

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

      9. lower-fma.f64 N/A

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

      10. distribute-lft-neg-in N/A

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

      11. *-commutative N/A

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

      12. distribute-rgt-neg-in N/A

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-expm1.f64 90.7

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

   10. Applied rewrites 90.7%

       \[\leadsto x - \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(y, t \cdot 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 93.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(y, t \cdot 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left(1 - y\right) + y \cdot e^{z} \leq 0:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \mathsf{expm1}\left(z\right) \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (- 1.0 y) (* y (exp z))) 0.0)
   (- x (/ (* y z) t))
   (- x (* (expm1 z) (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((1.0 - y) + (y * exp(z))) <= 0.0) {
		tmp = x - ((y * z) / t);
	} else {
		tmp = x - (expm1(z) * (y / t));
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (((1.0 - y) + (y * Math.exp(z))) <= 0.0) {
		tmp = x - ((y * z) / t);
	} else {
		tmp = x - (Math.expm1(z) * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if ((1.0 - y) + (y * math.exp(z))) <= 0.0:
		tmp = x - ((y * z) / t)
	else:
		tmp = x - (math.expm1(z) * (y / t))
	return tmp

Julia

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 0.0], N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(Exp[z] - 1), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 0.0

   1. Initial program 1.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 88.0%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-/.f64 87.7

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

   5. Applied rewrites 87.7%

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

Alternative 5: 75.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 0.0

   1. Initial program 1.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 80.2

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

   5. Applied rewrites 80.2%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 88.0%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. distribute-lft-out-- N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 78.3%

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

   9. Taylor expanded in z around 0

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

       1. +-commutative N/A

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

       2. mul-1-neg N/A

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

       3. *-commutative N/A

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

       4. associate-*r/ N/A

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

       5. distribute-rgt-neg-out N/A

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

       6. lower-fma.f64 N/A

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

       7. distribute-neg-frac2 N/A

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

       8. mul-1-neg N/A

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

       9. lower-/.f64 N/A

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

       10. mul-1-neg N/A

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

       11. lower-neg.f64 79.3

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

   11. Applied rewrites 79.3%

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

Alternative 6: 91.1% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.8 \cdot 10^{+42}:\\ \;\;\;\;x - \mathsf{expm1}\left(z\right) \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.8e+42) (- x (* (expm1 z) (/ y t))) (- x (/ (log1p (* y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.8e+42) {
		tmp = x - (expm1(z) * (y / t));
	} else {
		tmp = x - (log1p((y * z)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.8e+42) {
		tmp = x - (Math.expm1(z) * (y / t));
	} else {
		tmp = x - (Math.log1p((y * z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5.8e+42:
		tmp = x - (math.expm1(z) * (y / t))
	else:
		tmp = x - (math.log1p((y * z)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.8e+42)
		tmp = Float64(x - Float64(expm1(z) * Float64(y / t)));
	else
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.8e+42], N[(x - N[(N[(Exp[z] - 1), $MachinePrecision] * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.79999999999999961e42

   1. Initial program 84.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-expm1.f64 N/A

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

      5. lower-/.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -5.79999999999999961e42 < z

   1. Initial program 60.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. distribute-lft-out-- N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 98.5

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

   5. Applied rewrites 98.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 95.9%

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

Alternative 7: 82.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -0.036:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(y, 0.16666666666666666, y \cdot \left(z \cdot 0.041666666666666664\right)\right), y \cdot 0.5\right), y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -0.036)
   (- x (/ (log 1.0) t))
   (-
    x
    (/
     (*
      z
      (fma
       z
       (fma
        z
        (fma y 0.16666666666666666 (* y (* z 0.041666666666666664)))
        (* y 0.5))
       y))
     t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -0.036) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - ((z * fma(z, fma(z, fma(y, 0.16666666666666666, (y * (z * 0.041666666666666664))), (y * 0.5)), y)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -0.036)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(Float64(z * fma(z, fma(z, fma(y, 0.16666666666666666, Float64(y * Float64(z * 0.041666666666666664))), Float64(y * 0.5)), y)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -0.036], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z * N[(z * N[(z * N[(y * 0.16666666666666666 + N[(y * N[(z * 0.041666666666666664), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(y * 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -0.0359999999999999973

   1. Initial program 87.7%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\log \color{blue}{1}}{t} \]
   4. Step-by-step derivation

   5. Applied rewrites 60.2%

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

   if -0.0359999999999999973 < z

   1. Initial program 57.6%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

      4. cancel-sign-sub-inv N/A

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

      5. distribute-lft-out-- N/A

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 98.4

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

   5. Applied rewrites 98.4%

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

   6. Taylor expanded in y around 0

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

      1. lower-*.f64 N/A

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

      2. lower-expm1.f64 89.5

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

   8. Applied rewrites 89.5%

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

   9. Taylor expanded in z around 0

      \[\leadsto x - \frac{z \cdot \color{blue}{\left(y + z \cdot \left(\frac{1}{2} \cdot y + z \cdot \left(\frac{1}{24} \cdot \left(y \cdot z\right) + \frac{1}{6} \cdot y\right)\right)\right)}}{t} \]
   10. Step-by-step derivation

   11. Applied rewrites 89.7%

       \[\leadsto x - \frac{z \cdot \color{blue}{\mathsf{fma}\left(z, \mathsf{fma}\left(z, \mathsf{fma}\left(y, 0.16666666666666666, y \cdot \left(z \cdot 0.041666666666666664\right)\right), y \cdot 0.5\right), y\right)}}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 8: 76.3% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+210}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(-0.5, \frac{z \cdot \left(t \cdot \left(y - y \cdot y\right)\right)}{y \cdot y}, \frac{t}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -4.1e+210)
   (+
    x
    (/ -1.0 (/ (fma -0.5 (/ (* z (* t (- y (* y y)))) (* y y)) (/ t y)) z)))
   (- x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -4.1e+210) {
		tmp = x + (-1.0 / (fma(-0.5, ((z * (t * (y - (y * y)))) / (y * y)), (t / y)) / z));
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -4.1e+210)
		tmp = Float64(x + Float64(-1.0 / Float64(fma(-0.5, Float64(Float64(z * Float64(t * Float64(y - Float64(y * y)))) / Float64(y * y)), Float64(t / y)) / z)));
	else
		tmp = Float64(x - Float64(Float64(y * z) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -4.1e+210], N[(x + N[(-1.0 / N[(N[(-0.5 * N[(N[(z * N[(t * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -4.10000000000000001e210

   1. Initial program 93.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 17.6

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

   5. Applied rewrites 17.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 17.6

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

   7. Applied rewrites 17.6%

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

   8. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

   10. Applied rewrites 63.9%

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

   if -4.10000000000000001e210 < z

   1. Initial program 63.4%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 81.1

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

   5. Applied rewrites 81.1%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.1 \cdot 10^{+210}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(-0.5, \frac{z \cdot \left(t \cdot \left(y - y \cdot y\right)\right)}{y \cdot y}, \frac{t}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 72.0% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 65.1%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

   4. cancel-sign-sub-inv N/A

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

   5. distribute-lft-out-- N/A

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

   6. lower-log1p.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-expm1.f64 98.8

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

5. Applied rewrites 98.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 84.0%

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

9. Taylor expanded in z around 0

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

    1. +-commutative N/A

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

    2. mul-1-neg N/A

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

    3. *-commutative N/A

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

    4. associate-*r/ N/A

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

    5. distribute-rgt-neg-out N/A

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

    6. lower-fma.f64 N/A

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

    7. distribute-neg-frac2 N/A

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

    8. mul-1-neg N/A

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

    9. lower-/.f64 N/A

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

    10. mul-1-neg N/A

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

    11. lower-neg.f64 77.3

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

11. Applied rewrites 77.3%

    \[\leadsto \color{blue}{\mathsf{fma}\left(z, \frac{y}{-t}, x\right)} \]
12. Add Preprocessing

Alternative 10: 15.7% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 65.1%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
2. Add Preprocessing

3. Taylor expanded in x around 0

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

   1. mul-1-neg N/A

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

   2. distribute-neg-frac2 N/A

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

   3. lower-/.f64 N/A

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

   4. sub-neg N/A

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

   5. associate-+l+ N/A

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

   6. *-rgt-identity N/A

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

   7. cancel-sign-sub-inv N/A

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

   8. distribute-lft-out-- N/A

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

   9. lower-log1p.f64 N/A

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

   10. lower-*.f64 N/A

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

   11. lower-expm1.f64 N/A

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

   12. lower-neg.f64 27.7

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

5. Applied rewrites 27.7%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 17.5%

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

9. Taylor expanded in z around 0

   \[\leadsto \mathsf{neg}\left(y \cdot \frac{z}{t}\right) \]
10. Step-by-step derivation

11. Applied rewrites 14.6%

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

12. Final simplification 14.6%

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

Developer Target 1: 74.3% accurate, 1.8× 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 2024233 
(FPCore (x y z t)
  :name "System.Random.MWC.Distributions:truncatedExp from mwc-random-0.13.3.2"
  :precision binary64

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

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