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

Percentage Accurate: 61.7% → 98.5%

Time: 20.3s

Alternatives: 9

Speedup: 226.0×

• could not determine a ground truth

  1. x = 0.21183892812726624
  2. y = 1.9964944484512884e+70
  3. z = 4.4296048240340067e+46
  4. t = -1.0262025165933572e-43

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: 61.7% 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 64.1%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 99.4

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

5. Simplified 99.4%

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

Alternative 2: 93.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - y\right) + y \cdot e^{z}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 1:\\ \;\;\;\;x - \frac{y \cdot \mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{2}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- 1.0 y) (* y (exp z)))))
   (if (<= t_1 0.0)
     (- x (/ (log1p (* y z)) t))
     (if (<= t_1 1.0) (- x (/ (* y (expm1 z)) t)) (- x (/ 2.0 t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - y) + (y * exp(z));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - (log1p((y * z)) / t);
	} else if (t_1 <= 1.0) {
		tmp = x - ((y * expm1(z)) / t);
	} else {
		tmp = x - (2.0 / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (1.0 - y) + (y * Math.exp(z));
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - (Math.log1p((y * z)) / t);
	} else if (t_1 <= 1.0) {
		tmp = x - ((y * Math.expm1(z)) / t);
	} else {
		tmp = x - (2.0 / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (1.0 - y) + (y * math.exp(z))
	tmp = 0
	if t_1 <= 0.0:
		tmp = x - (math.log1p((y * z)) / t)
	elif t_1 <= 1.0:
		tmp = x - ((y * math.expm1(z)) / t)
	else:
		tmp = x - (2.0 / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(1.0 - y) + Float64(y * exp(z)))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	elseif (t_1 <= 1.0)
		tmp = Float64(x - Float64(Float64(y * expm1(z)) / t));
	else
		tmp = Float64(x - Float64(2.0 / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(1.0 - y), $MachinePrecision] + N[(y * N[Exp[z], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1.0], N[(x - N[(N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(2.0 / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 99.9

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

   5. Simplified 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

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

      1. lower-*.f64 99.9

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

   8. Simplified 99.9%

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

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

   1. Initial program 84.0%

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

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   4. 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 99.1

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

   5. Simplified 99.1%

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

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

   1. Initial program 98.7%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 98.8

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. associate-+l+ N/A

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

      16. lower-log1p.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 N/A

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

   4. Applied egg-rr 99.0%

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

      1. lift-exp.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-expm1.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.9

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

   6. Applied egg-rr 99.9%

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

   7. Taylor expanded in y around 0

      \[\leadsto x - \frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot \left(t \cdot y\right) + \frac{t}{e^{z} - 1}}{y}}} \]
   8. 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. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-expm1.f64 65.1

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

   9. Simplified 65.1%

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

   10. Taylor expanded in y around inf

       \[\leadsto x - \color{blue}{\frac{2}{t}} \]
   11. Step-by-step derivation

       1. lower-/.f64 65.1

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

   12. Simplified 65.1%

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

Alternative 3: 94.6% accurate, 0.9× speedup

Math

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

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 0.5 (* y t) (/ 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(0.5, (y * t), (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(0.5, Float64(y * t), 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[(0.5 * N[(y * t), $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 2.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 99.9

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

   5. Simplified 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

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

      1. lower-*.f64 99.9

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

   8. Simplified 99.9%

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

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

   1. Initial program 86.1%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 86.1

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. associate-+l+ N/A

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

      16. lower-log1p.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 N/A

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

   4. Applied egg-rr 94.2%

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

   5. Taylor expanded in y around 0

      \[\leadsto x - \frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot \left(t \cdot y\right) + \frac{t}{e^{z} - 1}}{y}}} \]
   6. 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. lower-fma.f64 N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 95.0

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

   7. Simplified 95.0%

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

4. Final simplification 96.2%

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

Alternative 4: 89.7% accurate, 1.0× speedup

Math

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

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 z (* 0.5 (- (* y t) t)) t) 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(z, (0.5 * ((y * t) - t)), t) / 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(Float64(fma(z, Float64(0.5 * Float64(Float64(y * t) - t)), t) / 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[(N[(z * N[(0.5 * N[(N[(y * t), $MachinePrecision] - t), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision] / z), $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 2.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 99.9

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

   5. Simplified 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

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

      1. lower-*.f64 99.9

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

   8. Simplified 99.9%

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

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

   1. Initial program 86.1%

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

      1. lift--.f64 N/A

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

      2. lift-exp.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-+.f64 N/A

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

      5. lift-log.f64 N/A

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

      6. remove-double-neg N/A

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

      7. remove-double-neg N/A

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

      8. clear-num N/A

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

      9. lower-/.f64 N/A

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

      10. lower-/.f64 86.1

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

      11. lift-log.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. associate-+l+ N/A

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

      16. lower-log1p.f64 N/A

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

      17. +-commutative N/A

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

      18. lift-*.f64 N/A

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

      19. lower-fma.f64 N/A

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

   4. Applied egg-rr 94.2%

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

      1. lift-exp.f64 N/A

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

      2. neg-mul-1 N/A

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

      3. *-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. metadata-eval N/A

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

      6. sub-neg N/A

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

      7. lift-exp.f64 N/A

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

      8. lift-expm1.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-*.f64 99.2

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

   6. Applied egg-rr 99.2%

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

   7. Taylor expanded in y around 0

      \[\leadsto x - \frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot \left(t \cdot y\right) + \frac{t}{e^{z} - 1}}{y}}} \]
   8. 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. lower-fma.f64 N/A

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

      9. *-commutative N/A

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

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

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

      11. metadata-eval N/A

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

      12. lower-*.f64 N/A

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

      13. lower-/.f64 N/A

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

      14. lower-expm1.f64 95.0

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

   9. Simplified 95.0%

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

   10. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. +-commutative N/A

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

       3. lower-fma.f64 N/A

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

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

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

       5. lower-*.f64 N/A

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

       6. lower--.f64 N/A

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

       7. *-commutative N/A

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

       8. lower-*.f64 87.9

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

   12. Simplified 87.9%

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

4. Final simplification 91.0%

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

Alternative 5: 85.1% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

   1. lift--.f64 N/A

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

   2. lift-exp.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-+.f64 N/A

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

   5. lift-log.f64 N/A

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

   6. remove-double-neg N/A

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

   7. remove-double-neg N/A

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

   8. clear-num N/A

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

   9. lower-/.f64 N/A

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

   10. lower-/.f64 64.1

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

   11. lift-log.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. lift--.f64 N/A

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

   14. sub-neg N/A

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

   15. associate-+l+ N/A

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

   16. lower-log1p.f64 N/A

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

   17. +-commutative N/A

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

   18. lift-*.f64 N/A

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

   19. lower-fma.f64 N/A

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

4. Applied egg-rr 87.1%

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

   1. lift-exp.f64 N/A

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

   2. neg-mul-1 N/A

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

   3. *-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. metadata-eval N/A

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

   6. sub-neg N/A

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

   7. lift-exp.f64 N/A

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

   8. lift-expm1.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-*.f64 99.4

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

6. Applied egg-rr 99.4%

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

7. Taylor expanded in y around 0

   \[\leadsto x - \frac{1}{\color{blue}{\frac{\frac{1}{2} \cdot \left(t \cdot y\right) + \frac{t}{e^{z} - 1}}{y}}} \]
8. 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. lower-fma.f64 N/A

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

   9. *-commutative N/A

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

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

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

   11. metadata-eval N/A

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

   12. lower-*.f64 N/A

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

   13. lower-/.f64 N/A

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

   14. lower-expm1.f64 92.1

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

9. Simplified 92.1%

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

10. Taylor expanded in z around 0

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

    1. lower-/.f64 N/A

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

    2. +-commutative N/A

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

    3. lower-fma.f64 N/A

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

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

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

    5. lower-*.f64 N/A

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

    6. lower--.f64 N/A

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

    7. *-commutative N/A

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

    8. lower-*.f64 86.9

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

12. Simplified 86.9%

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

13. Final simplification 86.9%

    \[\leadsto x + \frac{-1}{\frac{\frac{\mathsf{fma}\left(z, 0.5 \cdot \left(y \cdot t - t\right), t\right)}{z}}{y}} \]
14. Add Preprocessing

Alternative 6: 81.4% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{z}{t}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.3e-58) x (fma (- y) (/ z t) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.3e-58) {
		tmp = x;
	} else {
		tmp = fma(-y, (z / t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.3e-58)
		tmp = x;
	else
		tmp = fma(Float64(-y), Float64(z / t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.3e-58], x, N[((-y) * N[(z / t), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.30000000000000026e-58

   1. Initial program 79.1%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 68.9%

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

      1. metadata-eval N/A

         \[\leadsto x - \frac{\color{blue}{0}}{t} \]

      2. div0 N/A

         \[\leadsto x - \color{blue}{0} \]

      3. --rgt-identity 68.9

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

   7. Applied egg-rr 68.9%

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

   if -3.30000000000000026e-58 < z

   1. Initial program 56.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 93.1

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

   5. Simplified 93.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

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

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

      9. lift-neg.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 94.0

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

   7. Applied egg-rr 94.0%

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

Alternative 7: 80.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - \frac{y \cdot z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.3e-58) x (- x (/ (* y z) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.3e-58) {
		tmp = x;
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.3d-58)) then
        tmp = x
    else
        tmp = x - ((y * z) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.3e-58) {
		tmp = x;
	} else {
		tmp = x - ((y * z) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.3e-58:
		tmp = x
	else:
		tmp = x - ((y * z) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.3e-58)
		tmp = x;
	else
		tmp = Float64(x - Float64(Float64(y * z) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.3e-58)
		tmp = x;
	else
		tmp = x - ((y * z) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.3e-58], x, N[(x - N[(N[(y * z), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.30000000000000026e-58

   1. Initial program 79.1%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 68.9%

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

      1. metadata-eval N/A

         \[\leadsto x - \frac{\color{blue}{0}}{t} \]

      2. div0 N/A

         \[\leadsto x - \color{blue}{0} \]

      3. --rgt-identity 68.9

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

   7. Applied egg-rr 68.9%

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

   if -3.30000000000000026e-58 < z

   1. Initial program 56.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 93.1

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

   5. Simplified 93.1%

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

Alternative 8: 78.7% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-58}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;x - z \cdot \frac{y}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.3e-58) x (- x (* z (/ y t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.3e-58) {
		tmp = x;
	} else {
		tmp = x - (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) :: tmp
    if (z <= (-3.3d-58)) then
        tmp = x
    else
        tmp = x - (z * (y / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.3e-58) {
		tmp = x;
	} else {
		tmp = x - (z * (y / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.3e-58:
		tmp = x
	else:
		tmp = x - (z * (y / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.3e-58)
		tmp = x;
	else
		tmp = Float64(x - Float64(z * Float64(y / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.3e-58)
		tmp = x;
	else
		tmp = x - (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.3e-58], x, N[(x - N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.30000000000000026e-58

   1. Initial program 79.1%

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

   3. Taylor expanded in y around 0

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

   5. Simplified 68.9%

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

      1. metadata-eval N/A

         \[\leadsto x - \frac{\color{blue}{0}}{t} \]

      2. div0 N/A

         \[\leadsto x - \color{blue}{0} \]

      3. --rgt-identity 68.9

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

   7. Applied egg-rr 68.9%

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

   if -3.30000000000000026e-58 < z

   1. Initial program 56.1%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 93.1

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

   5. Simplified 93.1%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 86.3

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

   7. Applied egg-rr 86.3%

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

Alternative 9: 71.8% accurate, 226.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.1%

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

3. Taylor expanded in y around 0

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

5. Simplified 77.1%

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

   1. metadata-eval N/A

      \[\leadsto x - \frac{\color{blue}{0}}{t} \]

   2. div0 N/A

      \[\leadsto x - \color{blue}{0} \]

   3. --rgt-identity 77.1

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

7. Applied egg-rr 77.1%

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

Developer Target 1: 75.2% 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 2024207 
(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)))