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

Percentage Accurate: 61.3% → 98.2%

Time: 21.5s

Alternatives: 14

Speedup: 11.3×

• could not determine a ground truth

  1. x = -6.882131901677188e+48
  2. y = 1.5699019422306634e-124
  3. z = 1.5001067816108847e+60
  4. t = 4.261324207413454e-268

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: 61.3% 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: 98.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in z around inf

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

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

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

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

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

   6. lower-log1p.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-expm1.f64 98.4

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

5. Applied rewrites 98.4%

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

Alternative 2: 93.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(1 - y\right) + y \cdot e^{z}\\ \mathbf{if}\;t\_1 \leq 0:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \mathbf{elif}\;t\_1 \leq 500000000000:\\ \;\;\;\;\mathsf{fma}\left(-y, \frac{\mathsf{expm1}\left(z\right)}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\mathsf{log1p}\left(y \cdot \mathsf{expm1}\left(z\right)\right)}{-t}\\ \end{array} \end{array} \]

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 500000000000.0)
       (fma (- y) (/ (expm1 z) t) x)
       (/ (log1p (* y (expm1 z))) (- 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 <= 500000000000.0) {
		tmp = fma(-y, (expm1(z) / t), x);
	} else {
		tmp = log1p((y * expm1(z))) / -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 <= 500000000000.0)
		tmp = fma(Float64(-y), Float64(expm1(z) / t), x);
	else
		tmp = Float64(log1p(Float64(y * expm1(z))) / Float64(-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, 500000000000.0], N[((-y) * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], N[(N[Log[1 + N[(y * N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / (-t)), $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 2.0%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 5e11

   1. Initial program 83.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

   if 5e11 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 92.1%

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

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. sub-neg N/A

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

      5. associate-+l+ N/A

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

      6. *-rgt-identity N/A

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

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

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

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

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

      9. lower-log1p.f64 N/A

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

      10. lower-*.f64 N/A

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

      11. lower-expm1.f64 N/A

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

      12. lower-neg.f64 51.8

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

   5. Applied rewrites 51.8%

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

Alternative 3: 94.9% accurate, 0.6× speedup

Math

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

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)
       (fma (- y) (/ (expm1 z) t) x)
       (+ x (/ -1.0 (/ (fma 0.5 (* y t) (/ t (fma z (* z 0.5) z))) y)))))))

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 = fma(-y, (expm1(z) / t), x);
	} else {
		tmp = x + (-1.0 / (fma(0.5, (y * t), (t / fma(z, (z * 0.5), z))) / y));
	}
	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 = fma(Float64(-y), Float64(expm1(z) / t), x);
	else
		tmp = Float64(x + Float64(-1.0 / Float64(fma(0.5, Float64(y * t), Float64(t / fma(z, Float64(z * 0.5), z))) / y)));
	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[((-y) * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(-1.0 / N[(N[(0.5 * N[(y * t), $MachinePrecision] + N[(t / N[(z * N[(z * 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $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 2.0%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

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

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

      1. lift-expm1.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. sub-neg N/A

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

      5. +-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. lift-*.f64 N/A

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

      8. associate-/l* N/A

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

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

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 N/A

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

      12. lower-/.f64 99.9

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

   7. Applied rewrites 99.9%

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

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

   1. Initial program 88.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot y, y\right)}\right)}{t} \]

      8. *-commutative N/A

         \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{1}{2}}, y\right)\right)}{t} \]

      9. lower-*.f64 20.9

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

   8. Applied rewrites 20.9%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)}\right)}{t} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}{t} \]

      5. clear-num N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

      7. lower-/.f64 21.0

         \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)\right)}}} \]

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

          \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(y, z \cdot \frac{1}{2}, y\right)}\right)}} \]

      14. lower-*.f64 21.0

          \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, \color{blue}{z \cdot 0.5}, y\right)\right)}} \]

   10. Applied rewrites 21.0%

       \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, z \cdot 0.5, y\right)\right)}}} \]

   11. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. distribute-lft-in N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 53.5

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

   13. Applied rewrites 53.5%

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

4. Final simplification 96.3%

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

Alternative 4: 93.6% accurate, 0.6× speedup

Math

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

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 (/ -1.0 (/ (fma 0.5 (* y t) (/ t (fma z (* z 0.5) z))) y)))))))

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 + (-1.0 / (fma(0.5, (y * t), (t / fma(z, (z * 0.5), z))) / y));
	}
	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(-1.0 / Float64(fma(0.5, Float64(y * t), Float64(t / fma(z, Float64(z * 0.5), z))) / y)));
	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[(-1.0 / N[(N[(0.5 * N[(y * t), $MachinePrecision] + N[(t / N[(z * N[(z * 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $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 2.0%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 99.8

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

   8. Applied rewrites 99.8%

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

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

   1. Initial program 83.7%

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

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   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 98.3

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

   5. Applied rewrites 98.3%

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

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

   1. Initial program 88.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 95.1

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

   5. Applied rewrites 95.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot y, y\right)}\right)}{t} \]

      8. *-commutative N/A

         \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{1}{2}}, y\right)\right)}{t} \]

      9. lower-*.f64 20.9

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

   8. Applied rewrites 20.9%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)}\right)}{t} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}{t} \]

      5. clear-num N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

      7. lower-/.f64 21.0

         \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)\right)}}} \]

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

          \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(y, z \cdot \frac{1}{2}, y\right)}\right)}} \]

      14. lower-*.f64 21.0

          \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, \color{blue}{z \cdot 0.5}, y\right)\right)}} \]

   10. Applied rewrites 21.0%

       \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, z \cdot 0.5, y\right)\right)}}} \]

   11. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. distribute-lft-in N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 53.5

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

   13. Applied rewrites 53.5%

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

4. Final simplification 95.2%

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

Alternative 5: 87.9% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+23}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{log1p}\left(y \cdot z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6e+23) (- x (/ (log 1.0) t)) (- x (/ (log1p (* y z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6e+23) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - (log1p((y * z)) / t);
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6e+23) {
		tmp = x - (Math.log(1.0) / t);
	} else {
		tmp = x - (Math.log1p((y * z)) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -6e+23:
		tmp = x - (math.log(1.0) / t)
	else:
		tmp = x - (math.log1p((y * z)) / t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6e+23)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(log1p(Float64(y * z)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6e+23], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[1 + N[(y * z), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.0000000000000002e23

   1. Initial program 83.6%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 70.5%

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

   if -6.0000000000000002e23 < z

   1. Initial program 56.7%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 98.0

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

   5. Applied rewrites 98.0%

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

   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 96.8

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

   8. Applied rewrites 96.8%

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

Alternative 6: 82.7% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, z \cdot 0.5, z\right)\\ x + \frac{-1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(t \cdot t\_1\right) \cdot -0.08333333333333333, t \cdot 0.5\right), \frac{t}{t\_1}\right)}{y}} \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma z (* z 0.5) z)))
   (+
    x
    (/
     -1.0
     (/
      (fma y (fma y (* (* t t_1) -0.08333333333333333) (* t 0.5)) (/ t t_1))
      y)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(z, (z * 0.5), z);
	return x + (-1.0 / (fma(y, fma(y, ((t * t_1) * -0.08333333333333333), (t * 0.5)), (t / t_1)) / y));
}

Julia

function code(x, y, z, t)
	t_1 = fma(z, Float64(z * 0.5), z)
	return Float64(x + Float64(-1.0 / Float64(fma(y, fma(y, Float64(Float64(t * t_1) * -0.08333333333333333), Float64(t * 0.5)), Float64(t / t_1)) / y)))
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(z * 0.5), $MachinePrecision] + z), $MachinePrecision]}, N[(x + N[(-1.0 / N[(N[(y * N[(y * N[(N[(t * t$95$1), $MachinePrecision] * -0.08333333333333333), $MachinePrecision] + N[(t * 0.5), $MachinePrecision]), $MachinePrecision] + N[(t / t$95$1), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in z around inf

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

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

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

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

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

   6. lower-log1p.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-expm1.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot y, y\right)}\right)}{t} \]

   8. *-commutative N/A

      \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{1}{2}}, y\right)\right)}{t} \]

   9. lower-*.f64 79.9

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

8. Applied rewrites 79.9%

   \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)}\right)}{t} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log1p.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}{t} \]

   5. clear-num N/A

      \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

   6. lower-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

   7. lower-/.f64 79.9

      \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)\right)}}} \]

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

       \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(y, z \cdot \frac{1}{2}, y\right)}\right)}} \]

   14. lower-*.f64 79.9

       \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, \color{blue}{z \cdot 0.5}, y\right)\right)}} \]

10. Applied rewrites 79.9%

    \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, z \cdot 0.5, y\right)\right)}}} \]

11. Taylor expanded in y around 0

    \[\leadsto x - \frac{1}{\color{blue}{\frac{y \cdot \left(-1 \cdot \left(y \cdot \left(\frac{-1}{4} \cdot \left(t \cdot \left(z \cdot \left(1 + \frac{1}{2} \cdot z\right)\right)\right) + \frac{1}{3} \cdot \left(t \cdot \left(z \cdot \left(1 + \frac{1}{2} \cdot z\right)\right)\right)\right)\right) - \frac{-1}{2} \cdot t\right) + \frac{t}{z \cdot \left(1 + \frac{1}{2} \cdot z\right)}}{y}}} \]
12. Step-by-step derivation

    1. lower-/.f64 N/A

       \[\leadsto x - \frac{1}{\color{blue}{\frac{y \cdot \left(-1 \cdot \left(y \cdot \left(\frac{-1}{4} \cdot \left(t \cdot \left(z \cdot \left(1 + \frac{1}{2} \cdot z\right)\right)\right) + \frac{1}{3} \cdot \left(t \cdot \left(z \cdot \left(1 + \frac{1}{2} \cdot z\right)\right)\right)\right)\right) - \frac{-1}{2} \cdot t\right) + \frac{t}{z \cdot \left(1 + \frac{1}{2} \cdot z\right)}}{y}}} \]

13. Applied rewrites 84.7%

    \[\leadsto x - \frac{1}{\color{blue}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(t \cdot \mathsf{fma}\left(z, z \cdot 0.5, z\right)\right) \cdot -0.08333333333333333, t \cdot 0.5\right), \frac{t}{\mathsf{fma}\left(z, z \cdot 0.5, z\right)}\right)}{y}}} \]

14. Final simplification 84.7%

    \[\leadsto x + \frac{-1}{\frac{\mathsf{fma}\left(y, \mathsf{fma}\left(y, \left(t \cdot \mathsf{fma}\left(z, z \cdot 0.5, z\right)\right) \cdot -0.08333333333333333, t \cdot 0.5\right), \frac{t}{\mathsf{fma}\left(z, z \cdot 0.5, z\right)}\right)}{y}} \]
15. Add Preprocessing

Alternative 7: 80.9% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+62}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(-0.5, \frac{t \cdot \left(z \cdot \left(y - y \cdot y\right)\right)}{y \cdot y}, \frac{t}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, y \cdot t, \frac{t}{\mathsf{fma}\left(z, z \cdot 0.5, z\right)}\right)}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.6e+62)
   (+
    x
    (/ -1.0 (/ (fma -0.5 (/ (* t (* z (- y (* y y)))) (* y y)) (/ t y)) z)))
   (+ x (/ -1.0 (/ (fma 0.5 (* y t) (/ t (fma z (* z 0.5) z))) y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.6e+62) {
		tmp = x + (-1.0 / (fma(-0.5, ((t * (z * (y - (y * y)))) / (y * y)), (t / y)) / z));
	} else {
		tmp = x + (-1.0 / (fma(0.5, (y * t), (t / fma(z, (z * 0.5), z))) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.6e+62)
		tmp = Float64(x + Float64(-1.0 / Float64(fma(-0.5, Float64(Float64(t * Float64(z * Float64(y - Float64(y * y)))) / Float64(y * y)), Float64(t / y)) / z)));
	else
		tmp = Float64(x + Float64(-1.0 / Float64(fma(0.5, Float64(y * t), Float64(t / fma(z, Float64(z * 0.5), z))) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.6e+62], N[(x + N[(-1.0 / N[(N[(-0.5 * N[(N[(t * N[(z * N[(y - N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(y * y), $MachinePrecision]), $MachinePrecision] + N[(t / y), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 / N[(N[(0.5 * N[(y * t), $MachinePrecision] + N[(t / N[(z * N[(z * 0.5), $MachinePrecision] + z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.6e62

   1. Initial program 85.0%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot y, y\right)}\right)}{t} \]

      8. *-commutative N/A

         \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{1}{2}}, y\right)\right)}{t} \]

      9. lower-*.f64 22.7

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

   8. Applied rewrites 22.7%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)}\right)}{t} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}{t} \]

      5. clear-num N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

      7. lower-/.f64 22.7

         \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)\right)}}} \]

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

          \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(y, z \cdot \frac{1}{2}, y\right)}\right)}} \]

      14. lower-*.f64 22.7

          \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, \color{blue}{z \cdot 0.5}, y\right)\right)}} \]

   10. Applied rewrites 22.7%

       \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, z \cdot 0.5, y\right)\right)}}} \]

   11. Taylor expanded in z around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower-*.f64 N/A

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

       5. lower-*.f64 N/A

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

       6. mul-1-neg N/A

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

       7. unsub-neg N/A

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

       8. lower--.f64 N/A

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

       9. unpow2 N/A

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

       10. lower-*.f64 N/A

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

       11. unpow2 N/A

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

       12. lower-*.f64 N/A

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

       13. lower-/.f64 69.0

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

   13. Applied rewrites 69.0%

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

   if -3.6e62 < z

   1. Initial program 57.3%

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

   3. Taylor expanded in z around inf

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

      1. sub-neg N/A

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

      2. associate-+l+ N/A

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

      3. *-rgt-identity N/A

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

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

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

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

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

      6. lower-log1p.f64 N/A

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

      7. lower-*.f64 N/A

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

      8. lower-expm1.f64 98.1

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

   5. Applied rewrites 98.1%

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

   6. Taylor expanded in z around 0

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

      1. lower-*.f64 N/A

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

      2. +-commutative N/A

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

      3. *-commutative N/A

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

      4. *-commutative N/A

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

      5. associate-*r* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

         \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot y, y\right)}\right)}{t} \]

      8. *-commutative N/A

         \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{1}{2}}, y\right)\right)}{t} \]

      9. lower-*.f64 93.8

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

   8. Applied rewrites 93.8%

      \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)}\right)}{t} \]
   9. Step-by-step derivation

      1. lift-*.f64 N/A

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

      2. lift-fma.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift-log1p.f64 N/A

         \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}{t} \]

      5. clear-num N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

      6. lower-/.f64 N/A

         \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

      7. lower-/.f64 93.8

         \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)\right)}}} \]

      8. lift-fma.f64 N/A

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

      9. *-commutative N/A

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

      10. lift-*.f64 N/A

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

      11. associate-*l* N/A

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

      12. *-commutative N/A

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

      13. lower-fma.f64 N/A

          \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(y, z \cdot \frac{1}{2}, y\right)}\right)}} \]

      14. lower-*.f64 93.8

          \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, \color{blue}{z \cdot 0.5}, y\right)\right)}} \]

   10. Applied rewrites 93.8%

       \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, z \cdot 0.5, y\right)\right)}}} \]

   11. Taylor expanded in y around 0

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

       1. lower-/.f64 N/A

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

       2. lower-fma.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. lower-/.f64 N/A

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

       6. +-commutative N/A

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

       7. distribute-lft-in N/A

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

       8. *-rgt-identity N/A

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

       9. lower-fma.f64 N/A

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

       10. *-commutative N/A

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

       11. lower-*.f64 88.0

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

   13. Applied rewrites 88.0%

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

4. Final simplification 84.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+62}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(-0.5, \frac{t \cdot \left(z \cdot \left(y - y \cdot y\right)\right)}{y \cdot y}, \frac{t}{y}\right)}{z}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(0.5, y \cdot t, \frac{t}{\mathsf{fma}\left(z, z \cdot 0.5, z\right)}\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 79.6% accurate, 3.8× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in z around inf

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

   1. sub-neg N/A

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

   2. associate-+l+ N/A

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

   3. *-rgt-identity N/A

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

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

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

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

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

   6. lower-log1p.f64 N/A

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

   7. lower-*.f64 N/A

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

   8. lower-expm1.f64 98.4

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

5. Applied rewrites 98.4%

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

6. Taylor expanded in z around 0

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

   1. lower-*.f64 N/A

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

   2. +-commutative N/A

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

   3. *-commutative N/A

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

   4. *-commutative N/A

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

   5. associate-*r* N/A

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

   6. *-commutative N/A

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

   7. lower-fma.f64 N/A

      \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{2} \cdot y, y\right)}\right)}{t} \]

   8. *-commutative N/A

      \[\leadsto x - \frac{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, \color{blue}{y \cdot \frac{1}{2}}, y\right)\right)}{t} \]

   9. lower-*.f64 79.9

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

8. Applied rewrites 79.9%

   \[\leadsto x - \frac{\mathsf{log1p}\left(\color{blue}{z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)}\right)}{t} \]
9. Step-by-step derivation

   1. lift-*.f64 N/A

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

   2. lift-fma.f64 N/A

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

   3. lift-*.f64 N/A

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

   4. lift-log1p.f64 N/A

      \[\leadsto x - \frac{\color{blue}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}{t} \]

   5. clear-num N/A

      \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

   6. lower-/.f64 N/A

      \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot \frac{1}{2}, y\right)\right)}}} \]

   7. lower-/.f64 79.9

      \[\leadsto x - \frac{1}{\color{blue}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(z, y \cdot 0.5, y\right)\right)}}} \]

   8. lift-fma.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-*.f64 N/A

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

   11. associate-*l* N/A

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

   12. *-commutative N/A

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

   13. lower-fma.f64 N/A

       \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \color{blue}{\mathsf{fma}\left(y, z \cdot \frac{1}{2}, y\right)}\right)}} \]

   14. lower-*.f64 79.9

       \[\leadsto x - \frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, \color{blue}{z \cdot 0.5}, y\right)\right)}} \]

10. Applied rewrites 79.9%

    \[\leadsto x - \color{blue}{\frac{1}{\frac{t}{\mathsf{log1p}\left(z \cdot \mathsf{fma}\left(y, z \cdot 0.5, y\right)\right)}}} \]

11. Taylor expanded in y around 0

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

    1. lower-/.f64 N/A

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

    2. lower-fma.f64 N/A

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

    3. *-commutative N/A

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

    4. lower-*.f64 N/A

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

    5. lower-/.f64 N/A

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

    6. +-commutative N/A

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

    7. distribute-lft-in N/A

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

    8. *-rgt-identity N/A

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

    9. lower-fma.f64 N/A

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

    10. *-commutative N/A

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

    11. lower-*.f64 82.0

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

13. Applied rewrites 82.0%

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

14. Final simplification 82.0%

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

Alternative 9: 74.2% accurate, 7.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 77.6

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

5. Applied rewrites 77.6%

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

   1. lift-*.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 77.6

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

7. Applied rewrites 77.6%

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

8. Final simplification 77.6%

   \[\leadsto x + \frac{-1}{\frac{t}{y \cdot z}} \]
9. Add Preprocessing

Alternative 10: 74.2% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 77.6

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

5. Applied rewrites 77.6%

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

Alternative 11: 73.1% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 77.6

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

5. Applied rewrites 77.6%

   \[\leadsto x - \frac{\color{blue}{y \cdot z}}{t} \]
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 76.5

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

7. Applied rewrites 76.5%

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

Alternative 12: 15.2% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 77.6

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

5. Applied rewrites 77.6%

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

6. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. neg-mul-1 N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-*.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 14.4

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

8. Applied rewrites 14.4%

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

   1. lift-neg.f64 N/A

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

   2. associate-/l* N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 14.8

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

10. Applied rewrites 14.8%

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

11. Final simplification 14.8%

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

Alternative 13: 13.2% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

   \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
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 77.6

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

5. Applied rewrites 77.6%

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

6. Taylor expanded in x around 0

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

   1. associate-*r/ N/A

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

   2. lower-/.f64 N/A

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

   3. neg-mul-1 N/A

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

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

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

   5. mul-1-neg N/A

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

   6. lower-*.f64 N/A

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

   7. mul-1-neg N/A

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

   8. lower-neg.f64 14.4

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

8. Applied rewrites 14.4%

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

   1. lift-neg.f64 N/A

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

   2. *-commutative N/A

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

   3. associate-/l* N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 13.3

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

10. Applied rewrites 13.3%

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

11. Final simplification 13.3%

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

Alternative 14: 3.4% accurate, 13.3× speedup

Math

\[\begin{array}{l} \\ z \cdot \frac{-0.5}{t} \end{array} \]

FPCore

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

C

double code(double x, double y, double z, double t) {
	return z * (-0.5 / 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 = z * ((-0.5d0) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 62.7%

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

3. Taylor expanded in y around inf

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

   1. lower-+.f64 N/A

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

   2. lower-log.f64 N/A

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

   3. lower-expm1.f64 N/A

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

   4. mul-1-neg N/A

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

   5. log-rec N/A

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

   6. remove-double-neg N/A

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

   7. lower-log.f64 12.9

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

5. Applied rewrites 12.9%

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

6. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. lower-log.f64 12.6

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

8. Applied rewrites 12.6%

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

9. Taylor expanded in z around inf

   \[\leadsto \color{blue}{\frac{-1}{2} \cdot \frac{z}{t}} \]
10. Step-by-step derivation

    1. associate-*r/ N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot z}{t}} \]

    2. lower-/.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{2} \cdot z}{t}} \]

    3. *-commutative N/A

       \[\leadsto \frac{\color{blue}{z \cdot \frac{-1}{2}}}{t} \]

    4. lower-*.f64 3.5

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

11. Applied rewrites 3.5%

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

    1. associate-/l* N/A

       \[\leadsto \color{blue}{z \cdot \frac{\frac{-1}{2}}{t}} \]

    2. *-commutative N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{t} \cdot z} \]

    3. lower-*.f64 N/A

       \[\leadsto \color{blue}{\frac{\frac{-1}{2}}{t} \cdot z} \]

    4. lower-/.f64 3.5

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

13. Applied rewrites 3.5%

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

14. Final simplification 3.5%

    \[\leadsto z \cdot \frac{-0.5}{t} \]
15. Add Preprocessing

Developer Target 1: 74.9% 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 2024212 
(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)))