Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 96.8% → 99.7%

Time: 10.7s

Alternatives: 14

Speedup: 1.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.6%

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

   1. lift--.f64 N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-/.f64 N/A

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

   6. frac-2neg N/A

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

   7. associate-/r/ N/A

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

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

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

   9. remove-double-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift-+.f64 N/A

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

   13. +-commutative N/A

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

   14. distribute-neg-in N/A

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

   15. metadata-eval N/A

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

   16. unsub-neg N/A

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

   17. lower--.f64 99.9

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

4. Applied rewrites 99.9%

   \[\leadsto \color{blue}{\mathsf{fma}\left(\frac{y - z}{-1 - \left(t - z\right)}, a, x\right)} \]
5. Add Preprocessing

Alternative 2: 71.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \frac{y}{-1 - t}\\ t_2 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+302}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (/ y (- -1.0 t)))) (t_2 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_2 -1e+302)
     t_1
     (if (<= t_2 5e+162) (fma a (/ z (- 1.0 z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * (y / (-1.0 - t));
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -1e+302) {
		tmp = t_1;
	} else if (t_2 <= 5e+162) {
		tmp = fma(a, (z / (1.0 - z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * Float64(y / Float64(-1.0 - t)))
	t_2 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_2 <= -1e+302)
		tmp = t_1;
	elseif (t_2 <= 5e+162)
		tmp = fma(a, Float64(z / Float64(1.0 - z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+302], t$95$1, If[LessEqual[t$95$2, 5e+162], N[(a * N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -1.0000000000000001e302 or 4.9999999999999997e162 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 74.6

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 69.9%

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

   if -1.0000000000000001e302 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 4.9999999999999997e162

   1. Initial program 97.3%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.2%

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

Alternative 3: 65.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := a \cdot \frac{y}{-1 - t}\\ t_2 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+302}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+162}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* a (/ y (- -1.0 t)))) (t_2 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_2 -1e+302) t_1 (if (<= t_2 5e+162) (- x a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = a * (y / (-1.0 - t));
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -1e+302) {
		tmp = t_1;
	} else if (t_2 <= 5e+162) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = a * (y / ((-1.0d0) - t))
    t_2 = (y - z) / (((t - z) + 1.0d0) / a)
    if (t_2 <= (-1d+302)) then
        tmp = t_1
    else if (t_2 <= 5d+162) then
        tmp = x - a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = a * (y / (-1.0 - t));
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -1e+302) {
		tmp = t_1;
	} else if (t_2 <= 5e+162) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = a * (y / (-1.0 - t))
	t_2 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_2 <= -1e+302:
		tmp = t_1
	elif t_2 <= 5e+162:
		tmp = x - a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(a * Float64(y / Float64(-1.0 - t)))
	t_2 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_2 <= -1e+302)
		tmp = t_1;
	elseif (t_2 <= 5e+162)
		tmp = Float64(x - a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = a * (y / (-1.0 - t));
	t_2 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_2 <= -1e+302)
		tmp = t_1;
	elseif (t_2 <= 5e+162)
		tmp = x - a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+302], t$95$1, If[LessEqual[t$95$2, 5e+162], N[(x - a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -1.0000000000000001e302 or 4.9999999999999997e162 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in a around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. lower-+.f64 N/A

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

      9. distribute-lft-in N/A

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

      10. metadata-eval N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. lower--.f64 74.6

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

   7. Applied rewrites 74.6%

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

   8. Taylor expanded in z around 0

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

   10. Applied rewrites 69.9%

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

   if -1.0000000000000001e302 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 4.9999999999999997e162

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 66.7

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

   5. Applied rewrites 66.7%

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

Alternative 4: 61.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;\frac{y}{z} \cdot a\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+158}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{a}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_1 -2e+306)
     (* (/ y z) a)
     (if (<= t_1 2e+158) (- x a) (* y (/ a z))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = (y / z) * a;
	} else if (t_1 <= 2e+158) {
		tmp = x - a;
	} else {
		tmp = y * (a / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (y - z) / (((t - z) + 1.0d0) / a)
    if (t_1 <= (-2d+306)) then
        tmp = (y / z) * a
    else if (t_1 <= 2d+158) then
        tmp = x - a
    else
        tmp = y * (a / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_1 <= -2e+306) {
		tmp = (y / z) * a;
	} else if (t_1 <= 2e+158) {
		tmp = x - a;
	} else {
		tmp = y * (a / z);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_1 <= -2e+306:
		tmp = (y / z) * a
	elif t_1 <= 2e+158:
		tmp = x - a
	else:
		tmp = y * (a / z)
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_1 <= -2e+306)
		tmp = Float64(Float64(y / z) * a);
	elseif (t_1 <= 2e+158)
		tmp = Float64(x - a);
	else
		tmp = Float64(y * Float64(a / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_1 <= -2e+306)
		tmp = (y / z) * a;
	elseif (t_1 <= 2e+158)
		tmp = x - a;
	else
		tmp = y * (a / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e+306], N[(N[(y / z), $MachinePrecision] * a), $MachinePrecision], If[LessEqual[t$95$1, 2e+158], N[(x - a), $MachinePrecision], N[(y * N[(a / z), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -2.00000000000000003e306

   1. Initial program 100.0%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 91.5

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

   7. Applied rewrites 91.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto \frac{a \cdot y}{\color{blue}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 61.5%

       \[\leadsto \frac{a \cdot y}{\color{blue}{z}} \]
   11. Step-by-step derivation

   12. Applied rewrites 61.5%

       \[\leadsto \frac{y}{z} \cdot a \]

   if -2.00000000000000003e306 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.99999999999999991e158

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 66.8

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

   5. Applied rewrites 66.8%

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

   if 1.99999999999999991e158 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 57.7

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

   7. Applied rewrites 57.7%

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

   9. Applied rewrites 79.8%

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

   10. Taylor expanded in z around inf

       \[\leadsto y \cdot \frac{a}{\color{blue}{z}} \]
   11. Step-by-step derivation

   12. Applied rewrites 33.7%

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

Alternative 5: 61.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{z} \cdot a\\ t_2 := \frac{y - z}{\frac{\left(t - z\right) + 1}{a}}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+306}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+158}:\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y z) a)) (t_2 (/ (- y z) (/ (+ (- t z) 1.0) a))))
   (if (<= t_2 -2e+306) t_1 (if (<= t_2 2e+158) (- x a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / z) * a;
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = t_1;
	} else if (t_2 <= 2e+158) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = (y / z) * a
    t_2 = (y - z) / (((t - z) + 1.0d0) / a)
    if (t_2 <= (-2d+306)) then
        tmp = t_1
    else if (t_2 <= 2d+158) then
        tmp = x - a
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / z) * a;
	double t_2 = (y - z) / (((t - z) + 1.0) / a);
	double tmp;
	if (t_2 <= -2e+306) {
		tmp = t_1;
	} else if (t_2 <= 2e+158) {
		tmp = x - a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / z) * a
	t_2 = (y - z) / (((t - z) + 1.0) / a)
	tmp = 0
	if t_2 <= -2e+306:
		tmp = t_1
	elif t_2 <= 2e+158:
		tmp = x - a
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / z) * a)
	t_2 = Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a))
	tmp = 0.0
	if (t_2 <= -2e+306)
		tmp = t_1;
	elseif (t_2 <= 2e+158)
		tmp = Float64(x - a);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / z) * a;
	t_2 = (y - z) / (((t - z) + 1.0) / a);
	tmp = 0.0;
	if (t_2 <= -2e+306)
		tmp = t_1;
	elseif (t_2 <= 2e+158)
		tmp = x - a;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / z), $MachinePrecision] * a), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+306], t$95$1, If[LessEqual[t$95$2, 2e+158], N[(x - a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < -2.00000000000000003e306 or 1.99999999999999991e158 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. lower-+.f64 N/A

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

      6. distribute-lft-in N/A

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

      7. metadata-eval N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 67.4

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

   7. Applied rewrites 67.4%

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

   8. Taylor expanded in z around inf

      \[\leadsto \frac{a \cdot y}{\color{blue}{z}} \]
   9. Step-by-step derivation

   10. Applied rewrites 33.7%

       \[\leadsto \frac{a \cdot y}{\color{blue}{z}} \]
   11. Step-by-step derivation

   12. Applied rewrites 41.6%

       \[\leadsto \frac{y}{z} \cdot a \]

   if -2.00000000000000003e306 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.99999999999999991e158

   1. Initial program 97.3%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 66.8

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

   5. Applied rewrites 66.8%

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

Alternative 6: 92.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{-1 - t}, a, x\right)\\ \mathbf{if}\;t \leq -1000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.5 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z + -1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) (- -1.0 t)) a x)))
   (if (<= t -1000000000000.0)
     t_1
     (if (<= t 4.5e+59) (fma (/ (- y z) (+ z -1.0)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / (-1.0 - t)), a, x);
	double tmp;
	if (t <= -1000000000000.0) {
		tmp = t_1;
	} else if (t <= 4.5e+59) {
		tmp = fma(((y - z) / (z + -1.0)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / Float64(-1.0 - t)), a, x)
	tmp = 0.0
	if (t <= -1000000000000.0)
		tmp = t_1;
	elseif (t <= 4.5e+59)
		tmp = fma(Float64(Float64(y - z) / Float64(z + -1.0)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[t, -1000000000000.0], t$95$1, If[LessEqual[t, 4.5e+59], N[(N[(N[(y - z), $MachinePrecision] / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1e12 or 4.49999999999999959e59 < t

   1. Initial program 97.3%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around 0

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

      1. distribute-lft-in N/A

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

      2. metadata-eval N/A

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

      3. mul-1-neg N/A

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

      4. unsub-neg N/A

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

      5. lower--.f64 87.7

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

   7. Applied rewrites 87.7%

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

   if -1e12 < t < 4.49999999999999959e59

   1. Initial program 97.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 98.9

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

   7. Applied rewrites 98.9%

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

Alternative 7: 88.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1650000000000:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{t + \left(1 - z\right)}, x\right)\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+60}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{z + -1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - y \cdot \frac{a}{1 + t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1650000000000.0)
   (fma z (/ a (+ t (- 1.0 z))) x)
   (if (<= t 2e+60)
     (fma (/ (- y z) (+ z -1.0)) a x)
     (- x (* y (/ a (+ 1.0 t)))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1650000000000.0) {
		tmp = fma(z, (a / (t + (1.0 - z))), x);
	} else if (t <= 2e+60) {
		tmp = fma(((y - z) / (z + -1.0)), a, x);
	} else {
		tmp = x - (y * (a / (1.0 + t)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1650000000000.0)
		tmp = fma(z, Float64(a / Float64(t + Float64(1.0 - z))), x);
	elseif (t <= 2e+60)
		tmp = fma(Float64(Float64(y - z) / Float64(z + -1.0)), a, x);
	else
		tmp = Float64(x - Float64(y * Float64(a / Float64(1.0 + t))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1650000000000.0], N[(z * N[(a / N[(t + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 2e+60], N[(N[(N[(y - z), $MachinePrecision] / N[(z + -1.0), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - N[(y * N[(a / N[(1.0 + t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.65e12

   1. Initial program 96.9%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 82.5

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 82.5%

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

   if -1.65e12 < t < 1.9999999999999999e60

   1. Initial program 97.9%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in t around 0

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

      1. sub-neg N/A

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

      2. metadata-eval N/A

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

      3. lower-+.f64 98.9

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

   7. Applied rewrites 98.9%

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

   if 1.9999999999999999e60 < t

   1. Initial program 97.9%

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

   3. Taylor expanded in z around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-+.f64 85.8

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

   5. Applied rewrites 85.8%

      \[\leadsto x - \color{blue}{y \cdot \frac{a}{1 + t}} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 89.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \mathsf{fma}\left(\frac{a}{z}, 1 - y, a\right)\\ \mathbf{if}\;z \leq -600000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.7 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (fma (/ a z) (- 1.0 y) a))))
   (if (<= z -600000.0)
     t_1
     (if (<= z 3.7e+55) (fma (/ y (- -1.0 t)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - fma((a / z), (1.0 - y), a);
	double tmp;
	if (z <= -600000.0) {
		tmp = t_1;
	} else if (z <= 3.7e+55) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - fma(Float64(a / z), Float64(1.0 - y), a))
	tmp = 0.0
	if (z <= -600000.0)
		tmp = t_1;
	elseif (z <= 3.7e+55)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(a / z), $MachinePrecision] * N[(1.0 - y), $MachinePrecision] + a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -600000.0], t$95$1, If[LessEqual[z, 3.7e+55], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6e5 or 3.7000000000000002e55 < z

   1. Initial program 97.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

   5. Applied rewrites 87.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 89.6%

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

   if -6e5 < z < 3.7000000000000002e55

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\frac{y}{1 + t}\right)}, a, x\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{\mathsf{neg}\left(\left(1 + t\right)\right)}}, a, x\right) \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 89.1

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

   7. Applied rewrites 89.1%

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

Alternative 9: 88.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{z}{t + \left(1 - z\right)}, x\right)\\ \mathbf{if}\;z \leq -960000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+55}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ z (+ t (- 1.0 z))) x)))
   (if (<= z -960000.0)
     t_1
     (if (<= z 3.5e+55) (fma (/ y (- -1.0 t)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, (z / (t + (1.0 - z))), x);
	double tmp;
	if (z <= -960000.0) {
		tmp = t_1;
	} else if (z <= 3.5e+55) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(z / Float64(t + Float64(1.0 - z))), x)
	tmp = 0.0
	if (z <= -960000.0)
		tmp = t_1;
	elseif (z <= 3.5e+55)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(z / N[(t + N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -960000.0], t$95$1, If[LessEqual[z, 3.5e+55], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.6e5 or 3.5000000000000001e55 < z

   1. Initial program 97.5%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 87.4

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

   5. Applied rewrites 87.4%

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

   if -9.6e5 < z < 3.5000000000000001e55

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\frac{y}{1 + t}\right)}, a, x\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{\mathsf{neg}\left(\left(1 + t\right)\right)}}, a, x\right) \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 89.1

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

   7. Applied rewrites 89.1%

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

Alternative 10: 85.3% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1020000:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1 - z}, x\right)\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1020000.0)
   (fma a (/ z (- 1.0 z)) x)
   (if (<= z 1.1e+56) (fma (/ y (- -1.0 t)) a x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1020000.0) {
		tmp = fma(a, (z / (1.0 - z)), x);
	} else if (z <= 1.1e+56) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1020000.0)
		tmp = fma(a, Float64(z / Float64(1.0 - z)), x);
	elseif (z <= 1.1e+56)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1020000.0], N[(a * N[(z / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.1e+56], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.02e6

   1. Initial program 98.5%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 88.0

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 77.3%

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

   if -1.02e6 < z < 1.10000000000000008e56

   1. Initial program 97.8%

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

      1. lift--.f64 N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. associate-/r/ N/A

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

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

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. metadata-eval N/A

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

      16. unsub-neg N/A

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

      17. lower--.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\mathsf{neg}\left(\frac{y}{1 + t}\right)}, a, x\right) \]

      2. distribute-neg-frac2 N/A

         \[\leadsto \mathsf{fma}\left(\color{blue}{\frac{y}{\mathsf{neg}\left(\left(1 + t\right)\right)}}, a, x\right) \]

      3. mul-1-neg N/A

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

      4. lower-/.f64 N/A

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

      5. distribute-lft-in N/A

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

      6. metadata-eval N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. lower--.f64 89.1

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

   7. Applied rewrites 89.1%

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

   if 1.10000000000000008e56 < z

   1. Initial program 96.1%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 79.4

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

   5. Applied rewrites 79.4%

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

Alternative 11: 69.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.65 \cdot 10^{+38}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+148}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{1 + t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.65e+38)
   (- x a)
   (if (<= z 1.6e+148) (fma a (/ z (+ 1.0 t)) x) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.65e+38) {
		tmp = x - a;
	} else if (z <= 1.6e+148) {
		tmp = fma(a, (z / (1.0 + t)), x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.65e+38)
		tmp = Float64(x - a);
	elseif (z <= 1.6e+148)
		tmp = fma(a, Float64(z / Float64(1.0 + t)), x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.65e+38], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.6e+148], N[(a * N[(z / N[(1.0 + t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.65e38 or 1.6e148 < z

   1. Initial program 96.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 79.7

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

   5. Applied rewrites 79.7%

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

   if -1.65e38 < z < 1.6e148

   1. Initial program 98.1%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 63.3

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

   5. Applied rewrites 63.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 60.6%

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

Alternative 12: 64.1% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.2 \cdot 10^{+75}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.2e+75) (fma a (/ z t) x) (- x a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.2e+75) {
		tmp = fma(a, (z / t), x);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.2e+75)
		tmp = fma(a, Float64(z / t), x);
	else
		tmp = Float64(x - a);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.2e+75], N[(a * N[(z / t), $MachinePrecision] + x), $MachinePrecision], N[(x - a), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -1.2e75

   1. Initial program 96.0%

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

   3. Taylor expanded in y around 0

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

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

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

      2. metadata-eval N/A

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

      3. *-lft-identity N/A

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

      4. +-commutative N/A

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

      5. associate-/l* N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. +-commutative N/A

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

      9. associate--l+ N/A

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

      10. lower-+.f64 N/A

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

      11. lower--.f64 83.5

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 77.2%

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

   if -1.2e75 < t

   1. Initial program 98.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \color{blue}{x - a} \]
   4. Step-by-step derivation

      1. lower--.f64 60.5

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

   5. Applied rewrites 60.5%

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

Alternative 13: 61.7% accurate, 8.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x - a), $MachinePrecision]

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x - a} \]
4. Step-by-step derivation

   1. lower--.f64 58.6

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

5. Applied rewrites 58.6%

   \[\leadsto \color{blue}{x - a} \]
6. Add Preprocessing

Alternative 14: 16.8% accurate, 11.7× speedup

Math

\[\begin{array}{l} \\ -a \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := (-a)

Derivation

1. Initial program 97.6%

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

3. Taylor expanded in z around inf

   \[\leadsto \color{blue}{x - a} \]
4. Step-by-step derivation

   1. lower--.f64 58.6

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

5. Applied rewrites 58.6%

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

6. Taylor expanded in x around 0

   \[\leadsto -1 \cdot \color{blue}{a} \]
7. Step-by-step derivation

8. Applied rewrites 15.6%

   \[\leadsto -a \]
9. Add Preprocessing

Developer Target 1: 99.7% accurate, 1.2× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024254 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (/ (- y z) (+ (- t z) 1)) a)))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))