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

Percentage Accurate: 61.6% → 94.0%

Time: 21.8s

Alternatives: 16

Speedup: 11.3×

• could not determine a ground truth

  1. x = 9.915727709027043e-293
  2. y = 2.2203156756926244e-76
  3. z = 5.763010949097628e+263
  4. t = 6.838669372319712e-294

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: 61.6% 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: 94.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{\mathsf{log1p}\left(\mathsf{fma}\left(y, e^{z}, -y\right)\right)}}\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-126}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(y, t \cdot 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.6e-15)
   (+ x (/ -1.0 (/ t (log1p (fma y (exp z) (- y))))))
   (if (<= z -1.28e-126)
     (+ x (/ -1.0 (/ t (log (fma y z 1.0)))))
     (+ x (/ -1.0 (/ (fma y (* t 0.5) (/ t (expm1 z))) y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.6e-15) {
		tmp = x + (-1.0 / (t / log1p(fma(y, exp(z), -y))));
	} else if (z <= -1.28e-126) {
		tmp = x + (-1.0 / (t / log(fma(y, z, 1.0))));
	} else {
		tmp = x + (-1.0 / (fma(y, (t * 0.5), (t / expm1(z))) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.6e-15)
		tmp = Float64(x + Float64(-1.0 / Float64(t / log1p(fma(y, exp(z), Float64(-y))))));
	elseif (z <= -1.28e-126)
		tmp = Float64(x + Float64(-1.0 / Float64(t / log(fma(y, z, 1.0)))));
	else
		tmp = Float64(x + Float64(-1.0 / Float64(fma(y, Float64(t * 0.5), Float64(t / expm1(z))) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.6e-15], N[(x + N[(-1.0 / N[(t / N[Log[1 + N[(y * N[Exp[z], $MachinePrecision] + (-y)), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -1.28e-126], N[(x + N[(-1.0 / N[(t / N[Log[N[(y * z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 / N[(N[(y * N[(t * 0.5), $MachinePrecision] + N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.60000000000000004e-15

   1. Initial program 77.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 77.8

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

      5. lift-log.f64 N/A

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

      6. lift-+.f64 N/A

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

      7. lift--.f64 N/A

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

      8. sub-neg N/A

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

      9. associate-+l+ N/A

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

      10. lower-log1p.f64 N/A

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

      11. +-commutative N/A

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

      12. lift-*.f64 N/A

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

      13. lower-fma.f64 N/A

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

      14. lower-neg.f64 98.9

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

   4. Applied rewrites 98.9%

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

   if -2.60000000000000004e-15 < z < -1.28000000000000007e-126

   1. Initial program 55.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.1

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

   7. Applied rewrites 97.1%

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

   if -1.28000000000000007e-126 < z

   1. Initial program 49.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 91.2

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

   5. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 91.2

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

   7. Applied rewrites 91.2%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-expm1.f64 94.7

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

   10. Applied rewrites 94.7%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-15}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{\mathsf{log1p}\left(\mathsf{fma}\left(y, e^{z}, -y\right)\right)}}\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-126}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(y, t \cdot 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.7% accurate, 0.5× speedup

Math

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

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (- 1.0 y) (* y (exp z)))))
   (if (<= t_1 0.0)
     (fma (/ (log1p (* z y)) (* x t)) (- x) x)
     (if (<= t_1 2.0)
       (+ x (/ -1.0 (/ (fma y (* t 0.5) (/ t (expm1 z))) y)))
       (- x (/ (log t_1) 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 = fma((log1p((z * y)) / (x * t)), -x, x);
	} else if (t_1 <= 2.0) {
		tmp = x + (-1.0 / (fma(y, (t * 0.5), (t / expm1(z))) / y));
	} else {
		tmp = x - (log(t_1) / 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 = fma(Float64(log1p(Float64(z * y)) / Float64(x * t)), Float64(-x), x);
	elseif (t_1 <= 2.0)
		tmp = Float64(x + Float64(-1.0 / Float64(fma(y, Float64(t * 0.5), Float64(t / expm1(z))) / y)));
	else
		tmp = Float64(x - Float64(log(t_1) / 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[(N[(N[Log[1 + N[(z * y), $MachinePrecision]], $MachinePrecision] / N[(x * t), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision], If[LessEqual[t$95$1, 2.0], N[(x + N[(-1.0 / N[(N[(y * N[(t * 0.5), $MachinePrecision] + N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[t$95$1], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

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

   1. Initial program 2.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 88.7%

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

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

   1. Initial program 81.6%

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

   3. Taylor expanded in y around 0

      \[\leadsto x - \frac{\color{blue}{y \cdot \left(e^{z} - 1\right)}}{t} \]
   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 98.6

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

   5. Applied rewrites 98.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 98.6

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

   7. Applied rewrites 98.6%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-expm1.f64 99.9

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

   10. Applied rewrites 99.9%

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

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

   1. Initial program 94.2%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   2. Add Preprocessing
3. Recombined 3 regimes into one program.

4. Final simplification 95.9%

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

Alternative 3: 92.3% accurate, 0.9× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.1%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 88.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 88.7%

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

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

   1. Initial program 83.5%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 88.4

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

   5. Applied rewrites 88.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 88.3

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-expm1.f64 92.3

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

   10. Applied rewrites 92.3%

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

4. Final simplification 91.2%

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

Alternative 4: 94.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(\mathsf{fma}\left(y, e^{z}, -y\right)\right), x\right)\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-126}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(y, t \cdot 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.6e-15)
   (fma (/ -1.0 t) (log1p (fma y (exp z) (- y))) x)
   (if (<= z -1.28e-126)
     (+ x (/ -1.0 (/ t (log (fma y z 1.0)))))
     (+ x (/ -1.0 (/ (fma y (* t 0.5) (/ t (expm1 z))) y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.6e-15) {
		tmp = fma((-1.0 / t), log1p(fma(y, exp(z), -y)), x);
	} else if (z <= -1.28e-126) {
		tmp = x + (-1.0 / (t / log(fma(y, z, 1.0))));
	} else {
		tmp = x + (-1.0 / (fma(y, (t * 0.5), (t / expm1(z))) / y));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.6e-15)
		tmp = fma(Float64(-1.0 / t), log1p(fma(y, exp(z), Float64(-y))), x);
	elseif (z <= -1.28e-126)
		tmp = Float64(x + Float64(-1.0 / Float64(t / log(fma(y, z, 1.0)))));
	else
		tmp = Float64(x + Float64(-1.0 / Float64(fma(y, Float64(t * 0.5), Float64(t / expm1(z))) / y)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.6e-15], N[(N[(-1.0 / t), $MachinePrecision] * N[Log[1 + N[(y * N[Exp[z], $MachinePrecision] + (-y)), $MachinePrecision]], $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, -1.28e-126], N[(x + N[(-1.0 / N[(t / N[Log[N[(y * z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(-1.0 / N[(N[(y * N[(t * 0.5), $MachinePrecision] + N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.60000000000000004e-15

   1. Initial program 77.7%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. distribute-neg-frac2 N/A

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

      6. div-inv N/A

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

      7. *-commutative N/A

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

      8. lower-fma.f64 N/A

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

   4. Applied rewrites 98.7%

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

   if -2.60000000000000004e-15 < z < -1.28000000000000007e-126

   1. Initial program 55.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 97.1

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

   5. Applied rewrites 97.1%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 97.1

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

   7. Applied rewrites 97.1%

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

   if -1.28000000000000007e-126 < z

   1. Initial program 49.3%

      \[x - \frac{\log \left(\left(1 - y\right) + y \cdot e^{z}\right)}{t} \]
   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 91.2

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

   5. Applied rewrites 91.2%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 91.2

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

   7. Applied rewrites 91.2%

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

   8. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. metadata-eval N/A

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

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

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

      7. *-commutative N/A

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

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

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

      9. lower-fma.f64 N/A

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

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

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

      11. *-commutative N/A

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

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

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

      13. metadata-eval N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 N/A

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

      16. lower-expm1.f64 94.7

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

   10. Applied rewrites 94.7%

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

4. Final simplification 96.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{-1}{t}, \mathsf{log1p}\left(\mathsf{fma}\left(y, e^{z}, -y\right)\right), x\right)\\ \mathbf{elif}\;z \leq -1.28 \cdot 10^{-126}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;x + \frac{-1}{\frac{\mathsf{fma}\left(y, t \cdot 0.5, \frac{t}{\mathsf{expm1}\left(z\right)}\right)}{y}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+200}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), y\right), 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{log1p}\left(z \cdot y\right)}{x \cdot t}, -x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.5e+200)
   (-
    x
    (/ (log (fma z (fma z (* y (fma z 0.16666666666666666 0.5)) y) 1.0)) t))
   (if (<= y 1.2e+64)
     (- x (* y (/ (expm1 z) t)))
     (fma (/ (log1p (* z y)) (* x t)) (- x) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.5e+200) {
		tmp = x - (log(fma(z, fma(z, (y * fma(z, 0.16666666666666666, 0.5)), y), 1.0)) / t);
	} else if (y <= 1.2e+64) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = fma((log1p((z * y)) / (x * t)), -x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.5e+200)
		tmp = Float64(x - Float64(log(fma(z, fma(z, Float64(y * fma(z, 0.16666666666666666, 0.5)), y), 1.0)) / t));
	elseif (y <= 1.2e+64)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = fma(Float64(log1p(Float64(z * y)) / Float64(x * t)), Float64(-x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9.5e+200], N[(x - N[(N[Log[N[(z * N[(z * N[(y * N[(z * 0.16666666666666666 + 0.5), $MachinePrecision]), $MachinePrecision] + y), $MachinePrecision] + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+64], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[1 + N[(z * y), $MachinePrecision]], $MachinePrecision] / N[(x * t), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.49999999999999988e200

   1. Initial program 42.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right)\right) + 1\right)}}{t} \]

      2. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, y + z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right), 1\right)\right)}}{t} \]

      3. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \color{blue}{z \cdot \left(\frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y\right) + y}, 1\right)\right)}{t} \]

      4. lower-fma.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \color{blue}{\mathsf{fma}\left(z, \frac{1}{6} \cdot \left(y \cdot z\right) + \frac{1}{2} \cdot y, y\right)}, 1\right)\right)}{t} \]

      5. +-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{\frac{1}{2} \cdot y + \frac{1}{6} \cdot \left(y \cdot z\right)}, y\right), 1\right)\right)}{t} \]

      6. *-commutative N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{1}{2} \cdot y + \frac{1}{6} \cdot \color{blue}{\left(z \cdot y\right)}, y\right), 1\right)\right)}{t} \]

      7. associate-*r* N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{1}{2} \cdot y + \color{blue}{\left(\frac{1}{6} \cdot z\right) \cdot y}, y\right), 1\right)\right)}{t} \]

      8. distribute-rgt-out N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)}, y\right), 1\right)\right)}{t} \]

      9. lower-*.f64 N/A

         \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \color{blue}{y \cdot \left(\frac{1}{2} + \frac{1}{6} \cdot z\right)}, y\right), 1\right)\right)}{t} \]

      10. +-commutative N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y \cdot \color{blue}{\left(\frac{1}{6} \cdot z + \frac{1}{2}\right)}, y\right), 1\right)\right)}{t} \]

      11. *-commutative N/A

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y \cdot \left(\color{blue}{z \cdot \frac{1}{6}} + \frac{1}{2}\right), y\right), 1\right)\right)}{t} \]

      12. lower-fma.f64 63.7

          \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y \cdot \color{blue}{\mathsf{fma}\left(z, 0.16666666666666666, 0.5\right)}, y\right), 1\right)\right)}{t} \]

   5. Applied rewrites 63.7%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), y\right), 1\right)\right)}}{t} \]

   if -9.49999999999999988e200 < y < 1.2e64

   1. Initial program 68.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 1.2e64 < y

   1. Initial program 7.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 93.2%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+200}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, y \cdot \mathsf{fma}\left(z, 0.16666666666666666, 0.5\right), y\right), 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{log1p}\left(z \cdot y\right)}{x \cdot t}, -x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+200}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{log1p}\left(z \cdot y\right)}{x \cdot t}, -x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.5e+200)
   (+ x (/ -1.0 (/ t (log (fma y z 1.0)))))
   (if (<= y 1.2e+64)
     (- x (* y (/ (expm1 z) t)))
     (fma (/ (log1p (* z y)) (* x t)) (- x) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.5e+200) {
		tmp = x + (-1.0 / (t / log(fma(y, z, 1.0))));
	} else if (y <= 1.2e+64) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = fma((log1p((z * y)) / (x * t)), -x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.5e+200)
		tmp = Float64(x + Float64(-1.0 / Float64(t / log(fma(y, z, 1.0)))));
	elseif (y <= 1.2e+64)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = fma(Float64(log1p(Float64(z * y)) / Float64(x * t)), Float64(-x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9.5e+200], N[(x + N[(-1.0 / N[(t / N[Log[N[(y * z + 1.0), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+64], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[1 + N[(z * y), $MachinePrecision]], $MachinePrecision] / N[(x * t), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.49999999999999988e200

   1. Initial program 42.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 61.4

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

   5. Applied rewrites 61.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.4

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

   7. Applied rewrites 61.4%

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

   if -9.49999999999999988e200 < y < 1.2e64

   1. Initial program 68.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 1.2e64 < y

   1. Initial program 7.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 93.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+200}:\\ \;\;\;\;x + \frac{-1}{\frac{t}{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{log1p}\left(z \cdot y\right)}{x \cdot t}, -x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 89.7% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+200}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{log1p}\left(z \cdot y\right)}{x \cdot t}, -x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.5e+200)
   (- x (/ (log (fma y z 1.0)) t))
   (if (<= y 1.2e+64)
     (- x (* y (/ (expm1 z) t)))
     (fma (/ (log1p (* z y)) (* x t)) (- x) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.5e+200) {
		tmp = x - (log(fma(y, z, 1.0)) / t);
	} else if (y <= 1.2e+64) {
		tmp = x - (y * (expm1(z) / t));
	} else {
		tmp = fma((log1p((z * y)) / (x * t)), -x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.5e+200)
		tmp = Float64(x - Float64(log(fma(y, z, 1.0)) / t));
	elseif (y <= 1.2e+64)
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	else
		tmp = fma(Float64(log1p(Float64(z * y)) / Float64(x * t)), Float64(-x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9.5e+200], N[(x - N[(N[Log[N[(y * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 1.2e+64], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Log[1 + N[(z * y), $MachinePrecision]], $MachinePrecision] / N[(x * t), $MachinePrecision]), $MachinePrecision] * (-x) + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.49999999999999988e200

   1. Initial program 42.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if -9.49999999999999988e200 < y < 1.2e64

   1. Initial program 68.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 95.6

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

   5. Applied rewrites 95.6%

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

   if 1.2e64 < y

   1. Initial program 7.5%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-rgt-in N/A

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

      3. *-commutative N/A

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

      4. associate-*l* N/A

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

      5. *-lft-identity N/A

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

      6. lower-fma.f64 N/A

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

   5. Applied rewrites 89.8%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 93.2%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+200}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}{t}\\ \mathbf{elif}\;y \leq 1.2 \cdot 10^{+64}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{\mathsf{log1p}\left(z \cdot y\right)}{x \cdot t}, -x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 75.5% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((1.0d0 - y) + (y * exp(z))) <= 0.0d0) then
        tmp = x - ((z * y) * (1.0d0 / t))
    else
        tmp = x - (z * (y / t))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((1.0 - y) + (y * exp(z))) <= 0.0)
		tmp = x - ((z * y) * (1.0 / t));
	else
		tmp = x - (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 76.1

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 76.2%

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

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

   1. Initial program 83.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 75.7%

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

4. Final simplification 75.9%

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

Alternative 9: 75.6% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((1.0d0 - y) + (y * exp(z))) <= 0.0d0) then
        tmp = x - ((z * y) / t)
    else
        tmp = x - (z * (y / t))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((1.0 - y) + (y * exp(z))) <= 0.0)
		tmp = x - ((z * y) / t);
	else
		tmp = x - (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.1%

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

   3. Taylor expanded in z around 0

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

      1. lower-*.f64 76.2

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

   5. Applied rewrites 76.2%

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

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

   1. Initial program 83.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 75.7%

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

4. Final simplification 75.9%

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

Alternative 10: 75.4% accurate, 1.6× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (((1.0d0 - y) + (y * exp(z))) <= 0.0d0) then
        tmp = x - (y * (z / t))
    else
        tmp = x - (z * (y / t))
    end if
    code = tmp
end function

Java

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

Python

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

Julia

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

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (((1.0 - y) + (y * exp(z))) <= 0.0)
		tmp = x - (y * (z / t));
	else
		tmp = x - (z * (y / t));
	end
	tmp_2 = tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

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

   1. Initial program 2.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 76.1

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

   5. Applied rewrites 76.1%

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

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

   1. Initial program 83.5%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 73.0

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

   5. Applied rewrites 73.0%

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

   7. Applied rewrites 75.7%

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

Alternative 11: 87.5% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.5 \cdot 10^{+200}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(y, z, 1\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{\mathsf{expm1}\left(z\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.5e+200)
   (- x (/ (log (fma y z 1.0)) t))
   (- x (* y (/ (expm1 z) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.5e+200) {
		tmp = x - (log(fma(y, z, 1.0)) / t);
	} else {
		tmp = x - (y * (expm1(z) / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.5e+200)
		tmp = Float64(x - Float64(log(fma(y, z, 1.0)) / t));
	else
		tmp = Float64(x - Float64(y * Float64(expm1(z) / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9.5e+200], N[(x - N[(N[Log[N[(y * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -9.49999999999999988e200

   1. Initial program 42.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 61.4

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

   5. Applied rewrites 61.4%

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

   if -9.49999999999999988e200 < y

   1. Initial program 61.3%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 92.4

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

   5. Applied rewrites 92.4%

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

Alternative 12: 82.1% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-28}:\\ \;\;\;\;x - \frac{\log 1}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \mathsf{fma}\left(\mathsf{fma}\left(z, \mathsf{fma}\left(z, \frac{0.041666666666666664}{t}, \frac{0.16666666666666666}{t}\right), \frac{0.5}{t}\right), z \cdot z, \frac{z}{t}\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.05e-28)
   (- x (/ (log 1.0) t))
   (-
    x
    (*
     y
     (fma
      (fma
       z
       (fma z (/ 0.041666666666666664 t) (/ 0.16666666666666666 t))
       (/ 0.5 t))
      (* z z)
      (/ z t))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.05e-28) {
		tmp = x - (log(1.0) / t);
	} else {
		tmp = x - (y * fma(fma(z, fma(z, (0.041666666666666664 / t), (0.16666666666666666 / t)), (0.5 / t)), (z * z), (z / t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.05e-28)
		tmp = Float64(x - Float64(log(1.0) / t));
	else
		tmp = Float64(x - Float64(y * fma(fma(z, fma(z, Float64(0.041666666666666664 / t), Float64(0.16666666666666666 / t)), Float64(0.5 / t)), Float64(z * z), Float64(z / t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.05e-28], N[(x - N[(N[Log[1.0], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(N[(z * N[(z * N[(0.041666666666666664 / t), $MachinePrecision] + N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision] + N[(0.5 / t), $MachinePrecision]), $MachinePrecision] * N[(z * z), $MachinePrecision] + N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05000000000000003e-28

   1. Initial program 75.5%

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

   3. Taylor expanded in y around 0

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

   5. Applied rewrites 60.9%

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

   if -1.05000000000000003e-28 < z

   1. Initial program 50.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. lower-*.f64 N/A

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

      4. div-sub N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.6%

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

Alternative 13: 86.4% accurate, 1.9× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. div-sub N/A

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

   3. lower-*.f64 N/A

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

   4. div-sub N/A

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

   5. lower-/.f64 N/A

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

   6. lower-expm1.f64 85.4

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

5. Applied rewrites 85.4%

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

Alternative 14: 75.4% accurate, 4.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.5 \cdot 10^{+142}:\\ \;\;\;\;x - \frac{z \cdot \left(0.3333333333333333 \cdot \left(z \cdot \left(z \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.5e+142)
   (- x (/ (* z (* 0.3333333333333333 (* z (* z (* y (* y y)))))) t))
   (- x (* y (/ z t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.5e+142) {
		tmp = x - ((z * (0.3333333333333333 * (z * (z * (y * (y * y)))))) / t);
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-2.5d+142)) then
        tmp = x - ((z * (0.3333333333333333d0 * (z * (z * (y * (y * y)))))) / t)
    else
        tmp = x - (y * (z / t))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.5e+142) {
		tmp = x - ((z * (0.3333333333333333 * (z * (z * (y * (y * y)))))) / t);
	} else {
		tmp = x - (y * (z / t));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.5e+142:
		tmp = x - ((z * (0.3333333333333333 * (z * (z * (y * (y * y)))))) / t)
	else:
		tmp = x - (y * (z / t))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.5e+142)
		tmp = Float64(x - Float64(Float64(z * Float64(0.3333333333333333 * Float64(z * Float64(z * Float64(y * Float64(y * y)))))) / t));
	else
		tmp = Float64(x - Float64(y * Float64(z / t)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.5e+142)
		tmp = x - ((z * (0.3333333333333333 * (z * (z * (y * (y * y)))))) / t);
	else
		tmp = x - (y * (z / t));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.5e+142], N[(x - N[(N[(z * N[(0.3333333333333333 * N[(z * N[(z * N[(y * N[(y * y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(y * N[(z / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.5000000000000001e142

   1. Initial program 79.8%

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

   3. Taylor expanded in z around 0

      \[\leadsto x - \frac{\color{blue}{z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(z \cdot \left(y + \left(-3 \cdot {y}^{2} + 2 \cdot {y}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(y + -1 \cdot {y}^{2}\right)\right)\right)}}{t} \]
   4. Step-by-step derivation

      1. lower-*.f64 N/A

         \[\leadsto x - \frac{\color{blue}{z \cdot \left(y + z \cdot \left(\frac{1}{6} \cdot \left(z \cdot \left(y + \left(-3 \cdot {y}^{2} + 2 \cdot {y}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(y + -1 \cdot {y}^{2}\right)\right)\right)}}{t} \]

      2. +-commutative N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\left(z \cdot \left(\frac{1}{6} \cdot \left(z \cdot \left(y + \left(-3 \cdot {y}^{2} + 2 \cdot {y}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(y + -1 \cdot {y}^{2}\right)\right) + y\right)}}{t} \]

      3. lower-fma.f64 N/A

         \[\leadsto x - \frac{z \cdot \color{blue}{\mathsf{fma}\left(z, \frac{1}{6} \cdot \left(z \cdot \left(y + \left(-3 \cdot {y}^{2} + 2 \cdot {y}^{3}\right)\right)\right) + \frac{1}{2} \cdot \left(y + -1 \cdot {y}^{2}\right), y\right)}}{t} \]

   5. Applied rewrites 3.4%

      \[\leadsto x - \frac{\color{blue}{z \cdot \mathsf{fma}\left(z, \mathsf{fma}\left(0.5, y - y \cdot y, z \cdot \left(0.16666666666666666 \cdot \mathsf{fma}\left(y \cdot y, \mathsf{fma}\left(2, y, -3\right), y\right)\right)\right), y\right)}}{t} \]

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 55.7%

      \[\leadsto x - \frac{z \cdot \left(0.3333333333333333 \cdot \color{blue}{\left(z \cdot \left(z \cdot \left(y \cdot \left(y \cdot y\right)\right)\right)\right)}\right)}{t} \]

   if -2.5000000000000001e142 < z

   1. Initial program 55.3%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 80.7

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

   5. Applied rewrites 80.7%

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

Alternative 15: 74.3% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

3. Taylor expanded in z around 0

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

   1. associate-/l* N/A

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

   2. lower-*.f64 N/A

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

   3. lower-/.f64 74.0

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

5. Applied rewrites 74.0%

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

Alternative 16: 14.5% accurate, 11.9× speedup

Math

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

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(Float64(z * y) / Float64(-t))
end

MATLAB

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

Wolfram

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

Derivation

1. Initial program 59.0%

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

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

   7. *-rgt-identity N/A

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

   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 31.8

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

5. Applied rewrites 31.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 13.9%

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

9. Final simplification 13.9%

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

Developer Target 1: 74.3% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{-0.5}{y \cdot t}\\ \mathbf{if}\;z < -2.8874623088207947 \cdot 10^{+119}:\\ \;\;\;\;\left(x - \frac{t\_1}{z \cdot z}\right) - t\_1 \cdot \frac{\frac{2}{z}}{z \cdot z}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(1 + z \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- 0.5) (* y t))))
   (if (< z -2.8874623088207947e+119)
     (- (- x (/ t_1 (* z z))) (* t_1 (/ (/ 2.0 z) (* z z))))
     (- x (/ (log (+ 1.0 (* z y))) t)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = -0.5d0 / (y * t)
    if (z < (-2.8874623088207947d+119)) then
        tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0d0 / z) / (z * z)))
    else
        tmp = x - (log((1.0d0 + (z * y))) / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = -0.5 / (y * t);
	double tmp;
	if (z < -2.8874623088207947e+119) {
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	} else {
		tmp = x - (Math.log((1.0 + (z * y))) / t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = -0.5 / (y * t)
	tmp = 0
	if z < -2.8874623088207947e+119:
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)))
	else:
		tmp = x - (math.log((1.0 + (z * y))) / t)
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-0.5) / Float64(y * t))
	tmp = 0.0
	if (z < -2.8874623088207947e+119)
		tmp = Float64(Float64(x - Float64(t_1 / Float64(z * z))) - Float64(t_1 * Float64(Float64(2.0 / z) / Float64(z * z))));
	else
		tmp = Float64(x - Float64(log(Float64(1.0 + Float64(z * y))) / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = -0.5 / (y * t);
	tmp = 0.0;
	if (z < -2.8874623088207947e+119)
		tmp = (x - (t_1 / (z * z))) - (t_1 * ((2.0 / z) / (z * z)));
	else
		tmp = x - (log((1.0 + (z * y))) / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-0.5) / N[(y * t), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -2.8874623088207947e+119], N[(N[(x - N[(t$95$1 / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(t$95$1 * N[(N[(2.0 / z), $MachinePrecision] / N[(z * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(1.0 + N[(z * y), $MachinePrecision]), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024219 
(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)))