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

Percentage Accurate: 61.2% → 96.0%

Time: 18.7s

Alternatives: 15

Speedup: 11.3×

• could not determine a ground truth

  1. x = 1.2311254368640234e+58
  2. y = 1.0963499391060306e+133
  3. z = 6.643554078879203e+54
  4. t = 7.903967711184288e-32

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 61.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{expm1}\left(z\right) \cdot y\\ t_2 := e^{z} \cdot y + \left(1 - y\right)\\ \mathbf{if}\;t\_2 \leq 0:\\ \;\;\;\;\frac{t \cdot x - \mathsf{log1p}\left(t\_1\right)}{t}\\ \mathbf{elif}\;t\_2 \leq 1.000005:\\ \;\;\;\;x - \frac{1}{\mathsf{fma}\left(0.5, t, \frac{\frac{t}{\mathsf{expm1}\left(z\right)}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log t\_1}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (expm1 z) y)) (t_2 (+ (* (exp z) y) (- 1.0 y))))
   (if (<= t_2 0.0)
     (/ (- (* t x) (log1p t_1)) t)
     (if (<= t_2 1.000005)
       (- x (/ 1.0 (fma 0.5 t (/ (/ t (expm1 z)) y))))
       (- x (/ (log t_1) t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = expm1(z) * y;
	double t_2 = (exp(z) * y) + (1.0 - y);
	double tmp;
	if (t_2 <= 0.0) {
		tmp = ((t * x) - log1p(t_1)) / t;
	} else if (t_2 <= 1.000005) {
		tmp = x - (1.0 / fma(0.5, t, ((t / expm1(z)) / y)));
	} else {
		tmp = x - (log(t_1) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(expm1(z) * y)
	t_2 = Float64(Float64(exp(z) * y) + Float64(1.0 - y))
	tmp = 0.0
	if (t_2 <= 0.0)
		tmp = Float64(Float64(Float64(t * x) - log1p(t_1)) / t);
	elseif (t_2 <= 1.000005)
		tmp = Float64(x - Float64(1.0 / fma(0.5, t, Float64(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[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, 0.0], N[(N[(N[(t * x), $MachinePrecision] - N[Log[1 + t$95$1], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$2, 1.000005], N[(x - N[(1.0 / N[(0.5 * t + N[(N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $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 2.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-*r/ N/A

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

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

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. lower-neg.f64 N/A

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

      10. lower-/.f64 72.2

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

   5. Applied rewrites 72.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 18.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 N/A

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

       3. *-commutative N/A

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

       4. lower-*.f64 N/A

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

       5. associate--l+ N/A

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

       6. sub-neg N/A

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

       7. mul-1-neg N/A

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

       8. *-commutative N/A

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

       9. distribute-lft-in N/A

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

       10. metadata-eval N/A

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

       11. sub-neg N/A

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

       12. lower-log1p.f64 N/A

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

       13. *-commutative N/A

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

       14. lower-*.f64 N/A

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

       15. lower-expm1.f64 94.3

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

   11. Applied rewrites 94.3%

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

   if 0.0 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 1.00000500000000003

   1. Initial program 75.1%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 95.9

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

   5. Applied rewrites 95.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 95.8

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

   7. Applied rewrites 95.8%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 99.9

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

   10. Applied rewrites 99.9%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 99.9%

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

   if 1.00000500000000003 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 93.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 0:\\ \;\;\;\;\frac{t \cdot x - \mathsf{log1p}\left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \mathbf{elif}\;e^{z} \cdot y + \left(1 - y\right) \leq 1.000005:\\ \;\;\;\;x - \frac{1}{\mathsf{fma}\left(0.5, t, \frac{\frac{t}{\mathsf{expm1}\left(z\right)}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 94.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 1.000005:\\ \;\;\;\;x - \frac{1}{\mathsf{fma}\left(0.5, t, \frac{\frac{t}{\mathsf{expm1}\left(z\right)}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (+ (* (exp z) y) (- 1.0 y)) 1.000005)
   (- x (/ 1.0 (fma 0.5 t (/ (/ t (expm1 z)) y))))
   (- x (/ (log (* (expm1 z) y)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (((exp(z) * y) + (1.0 - y)) <= 1.000005) {
		tmp = x - (1.0 / fma(0.5, t, ((t / expm1(z)) / y)));
	} else {
		tmp = x - (log((expm1(z) * y)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(exp(z) * y) + Float64(1.0 - y)) <= 1.000005)
		tmp = Float64(x - Float64(1.0 / fma(0.5, t, Float64(Float64(t / expm1(z)) / y))));
	else
		tmp = Float64(x - Float64(log(Float64(expm1(z) * y)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[Exp[z], $MachinePrecision] * y), $MachinePrecision] + N[(1.0 - y), $MachinePrecision]), $MachinePrecision], 1.000005], N[(x - N[(1.0 / N[(0.5 * t + N[(N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(Exp[z] - 1), $MachinePrecision] * y), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z))) < 1.00000500000000003

   1. Initial program 52.5%

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

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 89.9

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

   5. Applied rewrites 89.9%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 89.8

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

   7. Applied rewrites 89.8%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 93.5

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

   10. Applied rewrites 93.5%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 94.4%

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

   if 1.00000500000000003 < (+.f64 (-.f64 #s(literal 1 binary64) y) (*.f64 y (exp.f64 z)))

   1. Initial program 93.8%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 96.9

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

   5. Applied rewrites 96.9%

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

4. Final simplification 94.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;e^{z} \cdot y + \left(1 - y\right) \leq 1.000005:\\ \;\;\;\;x - \frac{1}{\mathsf{fma}\left(0.5, t, \frac{\frac{t}{\mathsf{expm1}\left(z\right)}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{expm1}\left(z\right) \cdot y\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{\mathsf{fma}\left(0.5 \cdot z, t \cdot y - t, t\right)}{z}}{y}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.4e+18)
   (- x (/ 1.0 (/ (/ (fma (* 0.5 z) (- (* t y) t) t) z) y)))
   (if (<= y 4.5e+87)
     (- x (* (/ (expm1 z) t) y))
     (-
      x
      (/
       (log (fma (fma (* (fma 0.16666666666666666 z 0.5) y) z y) z 1.0))
       t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.4e+18) {
		tmp = x - (1.0 / ((fma((0.5 * z), ((t * y) - t), t) / z) / y));
	} else if (y <= 4.5e+87) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(fma((fma(0.16666666666666666, z, 0.5) * y), z, y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.4e+18)
		tmp = Float64(x - Float64(1.0 / Float64(Float64(fma(Float64(0.5 * z), Float64(Float64(t * y) - t), t) / z) / y)));
	elseif (y <= 4.5e+87)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(fma(Float64(fma(0.16666666666666666, z, 0.5) * y), z, y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.4e+18], N[(x - N[(1.0 / N[(N[(N[(N[(0.5 * z), $MachinePrecision] * N[(N[(t * y), $MachinePrecision] - t), $MachinePrecision] + t), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+87], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4e18

   1. Initial program 39.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 61.7

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

   5. Applied rewrites 61.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.7

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

   7. Applied rewrites 61.7%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 73.8

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

   10. Applied rewrites 73.8%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 76.1%

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

   if -5.4e18 < y < 4.5000000000000003e87

   1. Initial program 71.0%

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

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 4.5000000000000003e87 < y

   1. Initial program 13.4%

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

   3. Taylor expanded in z around 0

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 91.0

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

   5. Applied rewrites 91.0%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, y\right), z, 1\right)\right)}}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{\mathsf{fma}\left(0.5 \cdot z, t \cdot y - t, t\right)}{z}}{y}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.3% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{\mathsf{fma}\left(0.5 \cdot z, t \cdot y - t, t\right)}{z}}{y}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.4e+18)
   (- x (/ 1.0 (/ (/ (fma (* 0.5 z) (- (* t y) t) t) z) y)))
   (if (<= y 4.5e+87)
     (- x (* (/ (expm1 z) t) y))
     (- x (/ (log (fma (fma (* (* 0.16666666666666666 z) y) z y) z 1.0)) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.4e+18) {
		tmp = x - (1.0 / ((fma((0.5 * z), ((t * y) - t), t) / z) / y));
	} else if (y <= 4.5e+87) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(fma(((0.16666666666666666 * z) * y), z, y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.4e+18)
		tmp = Float64(x - Float64(1.0 / Float64(Float64(fma(Float64(0.5 * z), Float64(Float64(t * y) - t), t) / z) / y)));
	elseif (y <= 4.5e+87)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(fma(Float64(Float64(0.16666666666666666 * z) * y), z, y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.4e+18], N[(x - N[(1.0 / N[(N[(N[(N[(0.5 * z), $MachinePrecision] * N[(N[(t * y), $MachinePrecision] - t), $MachinePrecision] + t), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+87], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(N[(0.16666666666666666 * z), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4e18

   1. Initial program 39.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 61.7

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

   5. Applied rewrites 61.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.7

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

   7. Applied rewrites 61.7%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 73.8

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

   10. Applied rewrites 73.8%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 76.1%

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

   if -5.4e18 < y < 4.5000000000000003e87

   1. Initial program 71.0%

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

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 4.5000000000000003e87 < y

   1. Initial program 13.4%

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

   3. Taylor expanded in z around 0

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 91.0

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

   5. Applied rewrites 91.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 91.0%

      \[\leadsto x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \left(0.16666666666666666 \cdot z\right), z, y\right), z, 1\right)\right)}{t} \]
3. Recombined 3 regimes into one program.

4. Final simplification 92.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{\mathsf{fma}\left(0.5 \cdot z, t \cdot y - t, t\right)}{z}}{y}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\left(0.16666666666666666 \cdot z\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 90.9% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{1}{\mathsf{fma}\left(0.5, t, \frac{\frac{t}{\mathsf{expm1}\left(z\right)}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y 6.2e+87)
   (- x (/ 1.0 (fma 0.5 t (/ (/ t (expm1 z)) y))))
   (-
    x
    (/ (log (fma (fma (* (fma 0.16666666666666666 z 0.5) y) z y) z 1.0)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= 6.2e+87) {
		tmp = x - (1.0 / fma(0.5, t, ((t / expm1(z)) / y)));
	} else {
		tmp = x - (log(fma(fma((fma(0.16666666666666666, z, 0.5) * y), z, y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= 6.2e+87)
		tmp = Float64(x - Float64(1.0 / fma(0.5, t, Float64(Float64(t / expm1(z)) / y))));
	else
		tmp = Float64(x - Float64(log(fma(fma(Float64(fma(0.16666666666666666, z, 0.5) * y), z, y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, 6.2e+87], N[(x - N[(1.0 / N[(0.5 * t + N[(N[(t / N[(Exp[z] - 1), $MachinePrecision]), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(N[(N[(0.16666666666666666 * z + 0.5), $MachinePrecision] * y), $MachinePrecision] * z + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < 6.1999999999999999e87

   1. Initial program 61.8%

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

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 85.6

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

   5. Applied rewrites 85.6%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 85.6

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

   7. Applied rewrites 85.6%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 91.8

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

   10. Applied rewrites 91.8%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 92.7%

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

   if 6.1999999999999999e87 < y

   1. Initial program 13.4%

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

   3. Taylor expanded in z around 0

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

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. *-commutative N/A

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

      6. lower-fma.f64 N/A

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

      7. *-commutative N/A

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

      8. associate-*r* N/A

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

      9. distribute-rgt-out N/A

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

      10. lower-*.f64 N/A

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

      11. lower-fma.f64 91.0

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

   5. Applied rewrites 91.0%

      \[\leadsto x - \frac{\log \color{blue}{\left(\mathsf{fma}\left(\mathsf{fma}\left(y \cdot \mathsf{fma}\left(0.16666666666666666, z, 0.5\right), z, y\right), z, 1\right)\right)}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 92.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq 6.2 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{1}{\mathsf{fma}\left(0.5, t, \frac{\frac{t}{\mathsf{expm1}\left(z\right)}}{y}\right)}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(\mathsf{fma}\left(0.16666666666666666, z, 0.5\right) \cdot y, z, y\right), z, 1\right)\right)}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 90.3% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{\mathsf{fma}\left(0.5 \cdot z, t \cdot y - t, t\right)}{z}}{y}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(\mathsf{fma}\left(0.5, z \cdot y, y\right), z, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.4e+18)
   (- x (/ 1.0 (/ (/ (fma (* 0.5 z) (- (* t y) t) t) z) y)))
   (if (<= y 4.5e+87)
     (- x (* (/ (expm1 z) t) y))
     (- x (/ (log (fma (fma 0.5 (* z y) y) z 1.0)) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.4e+18) {
		tmp = x - (1.0 / ((fma((0.5 * z), ((t * y) - t), t) / z) / y));
	} else if (y <= 4.5e+87) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(fma(0.5, (z * y), y), z, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.4e+18)
		tmp = Float64(x - Float64(1.0 / Float64(Float64(fma(Float64(0.5 * z), Float64(Float64(t * y) - t), t) / z) / y)));
	elseif (y <= 4.5e+87)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(fma(0.5, Float64(z * y), y), z, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.4e+18], N[(x - N[(1.0 / N[(N[(N[(N[(0.5 * z), $MachinePrecision] * N[(N[(t * y), $MachinePrecision] - t), $MachinePrecision] + t), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+87], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(N[(0.5 * N[(z * y), $MachinePrecision] + y), $MachinePrecision] * z + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4e18

   1. Initial program 39.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 61.7

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

   5. Applied rewrites 61.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.7

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

   7. Applied rewrites 61.7%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 73.8

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

   10. Applied rewrites 73.8%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 76.1%

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

   if -5.4e18 < y < 4.5000000000000003e87

   1. Initial program 71.0%

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

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 4.5000000000000003e87 < y

   1. Initial program 13.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. +-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 91.0

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

   5. Applied rewrites 91.0%

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

Alternative 7: 90.3% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{\mathsf{fma}\left(0.5 \cdot z, t \cdot y - t, t\right)}{z}}{y}}\\ \mathbf{elif}\;y \leq 4.5 \cdot 10^{+87}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\log \left(\mathsf{fma}\left(z, y, 1\right)\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.4e+18)
   (- x (/ 1.0 (/ (/ (fma (* 0.5 z) (- (* t y) t) t) z) y)))
   (if (<= y 4.5e+87)
     (- x (* (/ (expm1 z) t) y))
     (- x (/ (log (fma z y 1.0)) t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.4e+18) {
		tmp = x - (1.0 / ((fma((0.5 * z), ((t * y) - t), t) / z) / y));
	} else if (y <= 4.5e+87) {
		tmp = x - ((expm1(z) / t) * y);
	} else {
		tmp = x - (log(fma(z, y, 1.0)) / t);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.4e+18)
		tmp = Float64(x - Float64(1.0 / Float64(Float64(fma(Float64(0.5 * z), Float64(Float64(t * y) - t), t) / z) / y)));
	elseif (y <= 4.5e+87)
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	else
		tmp = Float64(x - Float64(log(fma(z, y, 1.0)) / t));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.4e+18], N[(x - N[(1.0 / N[(N[(N[(N[(0.5 * z), $MachinePrecision] * N[(N[(t * y), $MachinePrecision] - t), $MachinePrecision] + t), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, 4.5e+87], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[Log[N[(z * y + 1.0), $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -5.4e18

   1. Initial program 39.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 61.7

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

   5. Applied rewrites 61.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.7

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

   7. Applied rewrites 61.7%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 73.8

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

   10. Applied rewrites 73.8%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 76.1%

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

   if -5.4e18 < y < 4.5000000000000003e87

   1. Initial program 71.0%

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

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 4.5000000000000003e87 < y

   1. Initial program 13.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 91.0

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

   5. Applied rewrites 91.0%

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

Alternative 8: 88.8% accurate, 1.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.4 \cdot 10^{+18}:\\ \;\;\;\;x - \frac{1}{\frac{\frac{\mathsf{fma}\left(0.5 \cdot z, t \cdot y - t, t\right)}{z}}{y}}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{\mathsf{expm1}\left(z\right)}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.4e+18)
   (- x (/ 1.0 (/ (/ (fma (* 0.5 z) (- (* t y) t) t) z) y)))
   (- x (* (/ (expm1 z) t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.4e+18) {
		tmp = x - (1.0 / ((fma((0.5 * z), ((t * y) - t), t) / z) / y));
	} else {
		tmp = x - ((expm1(z) / t) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.4e+18)
		tmp = Float64(x - Float64(1.0 / Float64(Float64(fma(Float64(0.5 * z), Float64(Float64(t * y) - t), t) / z) / y)));
	else
		tmp = Float64(x - Float64(Float64(expm1(z) / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.4e+18], N[(x - N[(1.0 / N[(N[(N[(N[(0.5 * z), $MachinePrecision] * N[(N[(t * y), $MachinePrecision] - t), $MachinePrecision] + t), $MachinePrecision] / z), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(N[(Exp[z] - 1), $MachinePrecision] / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -5.4e18

   1. Initial program 39.7%

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

   3. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 61.7

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

   5. Applied rewrites 61.7%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 61.7

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

   7. Applied rewrites 61.7%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 73.8

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

   10. Applied rewrites 73.8%

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

   11. Taylor expanded in z around 0

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

   13. Applied rewrites 76.1%

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

   if -5.4e18 < y

   1. Initial program 64.6%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 94.1

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

   5. Applied rewrites 94.1%

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

Alternative 9: 84.8% accurate, 4.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in y around 0

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

   1. *-commutative N/A

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

   2. lower-*.f64 N/A

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

   3. lower-expm1.f64 83.2

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

5. Applied rewrites 83.2%

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

   1. lift-/.f64 N/A

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

   2. clear-num N/A

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

   3. lower-/.f64 N/A

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

   4. lower-/.f64 83.2

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

7. Applied rewrites 83.2%

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

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. lower-fma.f64 N/A

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

   4. lower-*.f64 N/A

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

   5. lower-/.f64 N/A

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

   6. lower-expm1.f64 89.3

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

10. Applied rewrites 89.3%

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

11. Taylor expanded in z around 0

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

13. Applied rewrites 84.4%

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

Alternative 10: 81.6% accurate, 5.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.55 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{t} - \frac{1}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.55e+19) (fma (- (/ 1.0 t) (/ 1.0 t)) y x) (- x (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.55e+19) {
		tmp = fma(((1.0 / t) - (1.0 / t)), y, x);
	} else {
		tmp = x - ((z / t) * y);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.55e+19)
		tmp = fma(Float64(Float64(1.0 / t) - Float64(1.0 / t)), y, x);
	else
		tmp = Float64(x - Float64(Float64(z / t) * y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.55e+19], N[(N[(N[(1.0 / t), $MachinePrecision] - N[(1.0 / t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], N[(x - N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.55e19

   1. Initial program 79.2%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower-exp.f64 75.3

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

   5. Applied rewrites 75.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.2%

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

   if -3.55e19 < z

   1. Initial program 49.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.5%

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

Alternative 11: 78.3% accurate, 8.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+19}:\\ \;\;\;\;x - \frac{1}{0.5 \cdot t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{z}{t} \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5e+19) (- x (/ 1.0 (* 0.5 t))) (- x (* (/ z t) y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5e+19) {
		tmp = x - (1.0 / (0.5 * t));
	} else {
		tmp = x - ((z / t) * y);
	}
	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 <= (-5d+19)) then
        tmp = x - (1.0d0 / (0.5d0 * t))
    else
        tmp = x - ((z / t) * y)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5e+19) {
		tmp = x - (1.0 / (0.5 * t));
	} else {
		tmp = x - ((z / t) * y);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5e+19:
		tmp = x - (1.0 / (0.5 * t))
	else:
		tmp = x - ((z / t) * y)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5e+19)
		tmp = Float64(x - Float64(1.0 / Float64(0.5 * t)));
	else
		tmp = Float64(x - Float64(Float64(z / t) * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -5e+19)
		tmp = x - (1.0 / (0.5 * t));
	else
		tmp = x - ((z / t) * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5e+19], N[(x - N[(1.0 / N[(0.5 * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5e19

   1. Initial program 79.2%

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

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

      2. lower-*.f64 N/A

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

      3. lower-expm1.f64 75.4

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

   5. Applied rewrites 75.4%

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

      1. lift-/.f64 N/A

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

      2. clear-num N/A

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

      3. lower-/.f64 N/A

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

      4. lower-/.f64 75.3

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

   7. Applied rewrites 75.3%

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

   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. lower-fma.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-expm1.f64 86.1

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

   10. Applied rewrites 86.1%

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

   11. Taylor expanded in y around inf

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

   13. Applied rewrites 47.6%

       \[\leadsto x - \frac{1}{0.5 \cdot \color{blue}{t}} \]

   if -5e19 < z

   1. Initial program 49.2%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower-expm1.f64 89.5

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

   5. Applied rewrites 89.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 89.5%

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

Alternative 12: 74.0% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in y around 0

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

   1. associate-/l* N/A

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

   2. div-sub N/A

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

   3. *-commutative N/A

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

   4. lower-*.f64 N/A

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

   5. div-sub N/A

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

   6. lower-/.f64 N/A

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

   7. lower-expm1.f64 85.4

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

5. Applied rewrites 85.4%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 72.4%

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

Alternative 13: 72.5% accurate, 11.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r/ N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 71.7

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

5. Applied rewrites 71.7%

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

Alternative 14: 14.9% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r/ N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 71.7

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

5. Applied rewrites 71.7%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 15.3%

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

Alternative 15: 13.2% accurate, 11.9× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 57.9%

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

3. Taylor expanded in z around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-*r/ N/A

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

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

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

   6. mul-1-neg N/A

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

   7. lower-fma.f64 N/A

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

   8. mul-1-neg N/A

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

   9. lower-neg.f64 N/A

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

   10. lower-/.f64 71.7

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

5. Applied rewrites 71.7%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 15.3%

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

9. Taylor expanded in t around 0

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

11. Applied rewrites 14.7%

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

Developer Target 1: 74.7% 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 2024254 
(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)))