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

Percentage Accurate: 61.2% → 98.5%

Time: 17.9s

Alternatives: 12

Speedup: 11.3×

• could not determine a ground truth

  1. x = 1.1218143993671256e-72
  2. y = 2.673805341578034e+85
  3. z = 7.106346524639439e+160
  4. t = 9.476200327305497e+37

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: 61.2% 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

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 58.4%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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. *-commutative N/A

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

   8. lower-*.f64 N/A

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

   9. lower-expm1.f64 98.5

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

5. Applied rewrites 98.5%

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

Alternative 2: 95.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := e^{z} \cdot y + \left(1 - y\right)\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, z \cdot y, y\right) \cdot z\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 2:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(\left(\frac{y}{y \cdot y} - 1\right) \cdot \left(t \cdot z\right), -0.5, \frac{t}{y}\right)}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (* (exp z) y) (- 1.0 y))))
   (if (<= t_1 0.0)
     (- x (/ (log1p (* (fma 0.5 (* z y) y) z)) t))
     (if (<= t_1 2.0)
       (- x (* (/ (expm1 z) t) y))
       (-
        x
        (/ 1.0 (/ (fma (* (- (/ y (* y y)) 1.0) (* t z)) -0.5 (/ t y)) z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (exp(z) * y) + (1.0 - y);
	double tmp;
	if (t_1 <= 0.0) {
		tmp = x - (log1p((fma(0.5, (z * y), y) * z)) / t);
	} else if (t_1 <= 2.0) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (1.0 / (fma((((y / (y * y)) - 1.0) * (t * z)), -0.5, (t / y)) / z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(exp(z) * y) + Float64(1.0 - y))
	tmp = 0.0
	if (t_1 <= 0.0)
		tmp = Float64(x - Float64(log1p(Float64(fma(0.5, Float64(z * y), y) * z)) / t));
	elseif (t_1 <= 2.0)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(1.0 / Float64(fma(Float64(Float64(Float64(y / Float64(y * y)) - 1.0) * Float64(t * z)), -0.5, Float64(t / y)) / z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 0.0], N[(x - N[(N[Log[1 + N[(N[(0.5 * N[(z * y), $MachinePrecision] + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(1.0 / N[(N[(N[(N[(N[(y / N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $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 1.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-expm1.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + \frac{1}{2} \cdot \left(y \cdot z\right)\right)\right)}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, z \cdot y, y\right) \cdot z\right)}{t} \]

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

   1. Initial program 75.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 99.1

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

   5. Applied rewrites 99.1%

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

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

   1. Initial program 94.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 95.0

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-+l+ N/A

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

      10. lower-log1p.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 95.0

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

   4. Applied rewrites 95.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 55.6%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, z \cdot y, y\right) \cdot z\right)}{t}\\ \mathbf{elif}\;e^{z} \cdot y + \left(1 - y\right) \leq 2:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(\left(\frac{y}{y \cdot y} - 1\right) \cdot \left(t \cdot z\right), -0.5, \frac{t}{y}\right)}{z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 95.2% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 0.0], N[(x - N[(N[Log[1 + N[(N[(0.5 * N[(z * y), $MachinePrecision] + y), $MachinePrecision] * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(1.0 / N[(N[(N[(t * y), $MachinePrecision] * 0.5 + 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 1.8%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. lower-expm1.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

      \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \left(y + \frac{1}{2} \cdot \left(y \cdot z\right)\right)\right)}{t} \]
   7. Step-by-step derivation

   8. Applied rewrites 99.8%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\mathsf{fma}\left(0.5, z \cdot y, y\right) \cdot z\right)}{t} \]

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

   1. Initial program 80.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 80.2

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-+l+ N/A

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

      10. lower-log1p.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 87.5

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

   4. Applied rewrites 87.5%

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

   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. *-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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 89.8

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

   7. Applied rewrites 89.8%

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

4. Final simplification 92.6%

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

Alternative 4: 89.1% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+54}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(\left(\frac{y}{y \cdot y} - 1\right) \cdot \left(t \cdot z\right), -0.5, \frac{t}{y}\right)}{z}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+41}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.7e+54)
   (- x (/ 1.0 (/ (fma (* (- (/ y (* y y)) 1.0) (* t z)) -0.5 (/ t y)) z)))
   (if (<= y 4.2e+41)
     (- x (* (/ (expm1 z) t) y))
     (- x (/ (log (fma z y 1.0)) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.7e+54) {
		tmp = x - (1.0 / (fma((((y / (y * y)) - 1.0) * (t * z)), -0.5, (t / y)) / z));
	} else if (y <= 4.2e+41) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.7e+54)
		tmp = Float64(x - Float64(1.0 / Float64(fma(Float64(Float64(Float64(y / Float64(y * y)) - 1.0) * Float64(t * z)), -0.5, Float64(t / y)) / z)));
	elseif (y <= 4.2e+41)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.7e+54], N[(x - N[(1.0 / N[(N[(N[(N[(N[(y / N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.2e+41], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.69999999999999993e54

   1. Initial program 53.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 53.3

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-+l+ N/A

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

      10. lower-log1p.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 84.0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 65.5%

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

   if -4.69999999999999993e54 < y < 4.1999999999999999e41

   1. Initial program 70.9%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 96.5

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

   5. Applied rewrites 96.5%

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

   if 4.1999999999999999e41 < y

   1. Initial program 1.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 90.6

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

   5. Applied rewrites 90.6%

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

4. Final simplification 88.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+54}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(\left(\frac{y}{y \cdot y} - 1\right) \cdot \left(t \cdot z\right), -0.5, \frac{t}{y}\right)}{z}}\\ \mathbf{elif}\;y \leq 4.2 \cdot 10^{+41}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.2% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+54}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(\left(\frac{y}{y \cdot y} - 1\right) \cdot \left(t \cdot z\right), -0.5, \frac{t}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.7e+54)
   (- x (/ 1.0 (/ (fma (* (- (/ y (* y y)) 1.0) (* t z)) -0.5 (/ t y)) z)))
   (- x (* (/ (expm1 z) t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.7e+54) {
		tmp = x - (1.0 / (fma((((y / (y * y)) - 1.0) * (t * z)), -0.5, (t / y)) / z));
	} else {
		tmp = x - ((expm1(z) / t) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.7e+54)
		tmp = Float64(x - Float64(1.0 / Float64(fma(Float64(Float64(Float64(y / Float64(y * y)) - 1.0) * Float64(t * z)), -0.5, Float64(t / y)) / z)));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.7e+54], N[(x - N[(1.0 / N[(N[(N[(N[(N[(y / N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.69999999999999993e54

   1. Initial program 53.3%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 53.3

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-+l+ N/A

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

      10. lower-log1p.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 84.0

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

   4. Applied rewrites 84.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 65.5%

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

   if -4.69999999999999993e54 < y

   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 y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 91.5

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

   5. Applied rewrites 91.5%

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

4. Final simplification 85.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+54}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(\left(\frac{y}{y \cdot y} - 1\right) \cdot \left(t \cdot z\right), -0.5, \frac{t}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 83.7% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-83}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(\frac{-0.5}{y}, \frac{\left(\mathsf{fma}\left(-y, y, y\right) \cdot z\right) \cdot t}{y}, \frac{t}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\mathsf{fma}\left(\frac{z}{t}, 0.5, \frac{1}{t}\right) \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2e-83)
   (-
    x
    (/ 1.0 (/ (fma (/ -0.5 y) (/ (* (* (fma (- y) y y) z) t) y) (/ t y)) z)))
   (- x (* (* (fma (/ z t) 0.5 (/ 1.0 t)) z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2e-83) {
		tmp = x - (1.0 / (fma((-0.5 / y), (((fma(-y, y, y) * z) * t) / y), (t / y)) / z));
	} else {
		tmp = x - ((fma((z / t), 0.5, (1.0 / t)) * z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2e-83)
		tmp = Float64(x - Float64(1.0 / Float64(fma(Float64(-0.5 / y), Float64(Float64(Float64(fma(Float64(-y), y, y) * z) * t) / y), Float64(t / y)) / z)));
	else
		tmp = Float64(x - Float64(Float64(fma(Float64(z / t), 0.5, Float64(1.0 / t)) * z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2e-83], N[(x - N[(1.0 / N[(N[(N[(-0.5 / y), $MachinePrecision] * N[(N[(N[(N[((-y) * y + y), $MachinePrecision] * z), $MachinePrecision] * t), $MachinePrecision] / y), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(z / t), $MachinePrecision] * 0.5 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.0000000000000001e-83

   1. Initial program 79.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 79.8

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-+l+ N/A

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

      10. lower-log1p.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 96.6

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

   4. Applied rewrites 96.6%

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

      1. lift-fma.f64 N/A

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

      2. lift-neg.f64 N/A

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

      3. neg-mul-1 N/A

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

      4. distribute-rgt-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. lift-*.f64 99.9

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

   6. Applied rewrites 99.9%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

   9. Applied rewrites 70.5%

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

   if -2.0000000000000001e-83 < z

   1. Initial program 42.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.6%

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

Alternative 7: 83.6% accurate, 2.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.3 \cdot 10^{-82}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(\left(\frac{y}{y \cdot y} - 1\right) \cdot \left(t \cdot z\right), -0.5, \frac{t}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\mathsf{fma}\left(\frac{z}{t}, 0.5, \frac{1}{t}\right) \cdot z\right) \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -9.3e-82)
   (- x (/ 1.0 (/ (fma (* (- (/ y (* y y)) 1.0) (* t z)) -0.5 (/ t y)) z)))
   (- x (* (* (fma (/ z t) 0.5 (/ 1.0 t)) z) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -9.3e-82) {
		tmp = x - (1.0 / (fma((((y / (y * y)) - 1.0) * (t * z)), -0.5, (t / y)) / z));
	} else {
		tmp = x - ((fma((z / t), 0.5, (1.0 / t)) * z) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -9.3e-82)
		tmp = Float64(x - Float64(1.0 / Float64(fma(Float64(Float64(Float64(y / Float64(y * y)) - 1.0) * Float64(t * z)), -0.5, Float64(t / y)) / z)));
	else
		tmp = Float64(x - Float64(Float64(fma(Float64(z / t), 0.5, Float64(1.0 / t)) * z) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -9.3e-82], N[(x - N[(1.0 / N[(N[(N[(N[(N[(y / N[(y * y), $MachinePrecision]), $MachinePrecision] - 1.0), $MachinePrecision] * N[(t * z), $MachinePrecision]), $MachinePrecision] * -0.5 + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(N[(z / t), $MachinePrecision] * 0.5 + N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * z), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.3000000000000006e-82

   1. Initial program 79.8%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 79.8

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-+l+ N/A

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

      10. lower-log1p.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-fma.f64 N/A

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

      15. lower-neg.f64 96.6

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

   4. Applied rewrites 96.6%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

   7. Applied rewrites 70.3%

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

   if -9.3000000000000006e-82 < z

   1. Initial program 42.8%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 89.6%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.3 \cdot 10^{-82}:\\ \;\;\;\;x - \frac{1}{\frac{\mathsf{fma}\left(\left(\frac{y}{y \cdot y} - 1\right) \cdot \left(t \cdot z\right), -0.5, \frac{t}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x - \left(\mathsf{fma}\left(\frac{z}{t}, 0.5, \frac{1}{t}\right) \cdot z\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.2% accurate, 5.4× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 58.4

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

   5. lift-log.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. associate-+l+ N/A

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

   10. lower-log1p.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-neg.f64 80.7

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

4. Applied rewrites 80.7%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. associate-/r/ N/A

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

   4. lower-*.f64 N/A

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

6. Applied rewrites 98.2%

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

7. Taylor expanded in z around 0

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

   1. associate-/r* N/A

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

   2. lower-/.f64 N/A

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

   3. lower-/.f64 69.9

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

9. Applied rewrites 69.9%

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

Alternative 9: 74.3% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.4%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 58.4

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

   5. lift-log.f64 N/A

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

   6. lift-+.f64 N/A

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

   7. lift--.f64 N/A

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

   8. sub-neg N/A

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

   9. associate-+l+ N/A

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

   10. lower-log1p.f64 N/A

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

   11. +-commutative N/A

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

   12. lift-*.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower-neg.f64 80.7

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

4. Applied rewrites 80.7%

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

5. Taylor expanded in z around 0

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

   1. lower-/.f64 N/A

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

   2. *-commutative N/A

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

   3. lower-*.f64 69.8

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

7. Applied rewrites 69.8%

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

Alternative 10: 74.3% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 58.4%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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. *-commutative N/A

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

   2. lower-*.f64 69.8

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

5. Applied rewrites 69.8%

   \[\leadsto x - \frac{\color{blue}{z \cdot y}}{t} \]
6. Add Preprocessing

Alternative 11: 72.6% 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(Float64(-z), Float64(y / t), x)
end

Wolfram

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

Derivation

1. Initial program 58.4%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot z}{t}} \]
4. 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-lft-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 68.4

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

5. Applied rewrites 68.4%

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

Alternative 12: 14.5% accurate, 11.9× speedup

Math

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

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(Float64(-z) / t) * 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 58.4%

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

3. Taylor expanded in z around 0

   \[\leadsto \color{blue}{x + -1 \cdot \frac{y \cdot z}{t}} \]
4. 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-lft-neg-in N/A

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 68.4

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

5. Applied rewrites 68.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 16.0%

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

9. Final simplification 16.0%

   \[\leadsto \frac{-z}{t} \cdot y \]
10. Add Preprocessing

Developer Target 1: 75.5% 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 2024331 
(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)))