Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3

Percentage Accurate: 68.3% → 88.6%

Time: 9.2s

Alternatives: 16

Speedup: 0.8×

Specification

Math

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

FPCore

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

C

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

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 - x)) / (a - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - x)) / (a - z))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 88.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{z}\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \mathsf{fma}\left(-1, y, a\right), t\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z} + 1, t\_1 \cdot \mathsf{fma}\left(-1, y, a\right), t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) z)))
   (if (<= z -4.6e+114)
     (fma t_1 (fma -1.0 y a) t)
     (if (<= z 5.5e+170)
       (fma (/ (- y z) (- a z)) (- t x) x)
       (fma (+ (/ a z) 1.0) (* t_1 (fma -1.0 y a)) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) / z;
	double tmp;
	if (z <= -4.6e+114) {
		tmp = fma(t_1, fma(-1.0, y, a), t);
	} else if (z <= 5.5e+170) {
		tmp = fma(((y - z) / (a - z)), (t - x), x);
	} else {
		tmp = fma(((a / z) + 1.0), (t_1 * fma(-1.0, y, a)), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) / z)
	tmp = 0.0
	if (z <= -4.6e+114)
		tmp = fma(t_1, fma(-1.0, y, a), t);
	elseif (z <= 5.5e+170)
		tmp = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x);
	else
		tmp = fma(Float64(Float64(a / z) + 1.0), Float64(t_1 * fma(-1.0, y, a)), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -4.6e+114], N[(t$95$1 * N[(-1.0 * y + a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 5.5e+170], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(a / z), $MachinePrecision] + 1.0), $MachinePrecision] * N[(t$95$1 * N[(-1.0 * y + a), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.6000000000000001e114

   1. Initial program 35.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. div-sub N/A

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

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

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

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

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 90.2%

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

   if -4.6000000000000001e114 < z < 5.4999999999999999e170

   1. Initial program 83.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 89.5

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

      4. lift-fma.f64 N/A

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

   6. Applied rewrites 92.7%

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

   if 5.4999999999999999e170 < z

   1. Initial program 12.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 93.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, \mathsf{fma}\left(-1, y, a\right), t\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{z} + 1, \frac{t - x}{z} \cdot \mathsf{fma}\left(-1, y, a\right), t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 39.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a} \cdot y\\ t_2 := \mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-90}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.42 \cdot 10^{+19}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+99}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- t x) a) y)) (t_2 (fma x (/ z a) x)))
   (if (<= a -8.5e-6)
     t_2
     (if (<= a -3e-90)
       t_1
       (if (<= a 1.42e+19) t (if (<= a 9.6e+99) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((t - x) / a) * y;
	double t_2 = fma(x, (z / a), x);
	double tmp;
	if (a <= -8.5e-6) {
		tmp = t_2;
	} else if (a <= -3e-90) {
		tmp = t_1;
	} else if (a <= 1.42e+19) {
		tmp = t;
	} else if (a <= 9.6e+99) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(t - x) / a) * y)
	t_2 = fma(x, Float64(z / a), x)
	tmp = 0.0
	if (a <= -8.5e-6)
		tmp = t_2;
	elseif (a <= -3e-90)
		tmp = t_1;
	elseif (a <= 1.42e+19)
		tmp = t;
	elseif (a <= 9.6e+99)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -8.5e-6], t$95$2, If[LessEqual[a, -3e-90], t$95$1, If[LessEqual[a, 1.42e+19], t, If[LessEqual[a, 9.6e+99], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if a < -8.4999999999999999e-6 or 9.6000000000000005e99 < a

   1. Initial program 71.0%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 77.7

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

   5. Applied rewrites 77.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 60.6%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 55.4%

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

   if -8.4999999999999999e-6 < a < -3.0000000000000002e-90 or 1.42e19 < a < 9.6000000000000005e99

   1. Initial program 72.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 50.9%

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

   if -3.0000000000000002e-90 < a < 1.42e19

   1. Initial program 62.6%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 67.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 22.7

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

   7. Applied rewrites 22.7%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 45.8%

       \[\leadsto t \]
3. Recombined 3 regimes into one program.

4. Final simplification 50.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -8.5 \cdot 10^{-6}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{elif}\;a \leq -3 \cdot 10^{-90}:\\ \;\;\;\;\frac{t - x}{a} \cdot y\\ \mathbf{elif}\;a \leq 1.42 \cdot 10^{+19}:\\ \;\;\;\;t\\ \mathbf{elif}\;a \leq 9.6 \cdot 10^{+99}:\\ \;\;\;\;\frac{t - x}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{z}\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \mathsf{fma}\left(-1, y, a\right), t\right)\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-76}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+74}:\\ \;\;\;\;x + \frac{y - z}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t - t\_1 \cdot \left(y - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) z)))
   (if (<= z -6.5e+15)
     (fma t_1 (fma -1.0 y a) t)
     (if (<= z -2.2e-76)
       (+ x (/ (* (- y z) t) (- a z)))
       (if (<= z 3.5e+74)
         (+ x (* (/ (- y z) a) (- t x)))
         (- t (* t_1 (- y a))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) / z;
	double tmp;
	if (z <= -6.5e+15) {
		tmp = fma(t_1, fma(-1.0, y, a), t);
	} else if (z <= -2.2e-76) {
		tmp = x + (((y - z) * t) / (a - z));
	} else if (z <= 3.5e+74) {
		tmp = x + (((y - z) / a) * (t - x));
	} else {
		tmp = t - (t_1 * (y - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) / z)
	tmp = 0.0
	if (z <= -6.5e+15)
		tmp = fma(t_1, fma(-1.0, y, a), t);
	elseif (z <= -2.2e-76)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)));
	elseif (z <= 3.5e+74)
		tmp = Float64(x + Float64(Float64(Float64(y - z) / a) * Float64(t - x)));
	else
		tmp = Float64(t - Float64(t_1 * Float64(y - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -6.5e+15], N[(t$95$1 * N[(-1.0 * y + a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, -2.2e-76], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e+74], N[(x + N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t - N[(t$95$1 * N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.5e15

   1. Initial program 49.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. div-sub N/A

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

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

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

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

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 84.8%

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

   if -6.5e15 < z < -2.19999999999999999e-76

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -2.19999999999999999e-76 < z < 3.50000000000000014e74

   1. Initial program 87.3%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 77.4

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

   5. Applied rewrites 77.4%

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

   if 3.50000000000000014e74 < z

   1. Initial program 29.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 57.9

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

   4. Applied rewrites 57.9%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. distribute-rgt-out-- N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 84.3

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

   7. Applied rewrites 84.3%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, \mathsf{fma}\left(-1, y, a\right), t\right)\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-76}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+74}:\\ \;\;\;\;x + \frac{y - z}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \mathbf{if}\;z \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-76}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+74}:\\ \;\;\;\;x + \frac{y - z}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ (- t x) z) (- y a)))))
   (if (<= z -6.5e+15)
     t_1
     (if (<= z -2.2e-76)
       (+ x (/ (* (- y z) t) (- a z)))
       (if (<= z 3.5e+74) (+ x (* (/ (- y z) a) (- t x))) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - (((t - x) / z) * (y - a));
	double tmp;
	if (z <= -6.5e+15) {
		tmp = t_1;
	} else if (z <= -2.2e-76) {
		tmp = x + (((y - z) * t) / (a - z));
	} else if (z <= 3.5e+74) {
		tmp = x + (((y - z) / a) * (t - x));
	} 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) :: tmp
    t_1 = t - (((t - x) / z) * (y - a))
    if (z <= (-6.5d+15)) then
        tmp = t_1
    else if (z <= (-2.2d-76)) then
        tmp = x + (((y - z) * t) / (a - z))
    else if (z <= 3.5d+74) then
        tmp = x + (((y - z) / a) * (t - x))
    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 = t - (((t - x) / z) * (y - a));
	double tmp;
	if (z <= -6.5e+15) {
		tmp = t_1;
	} else if (z <= -2.2e-76) {
		tmp = x + (((y - z) * t) / (a - z));
	} else if (z <= 3.5e+74) {
		tmp = x + (((y - z) / a) * (t - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - (((t - x) / z) * (y - a))
	tmp = 0
	if z <= -6.5e+15:
		tmp = t_1
	elif z <= -2.2e-76:
		tmp = x + (((y - z) * t) / (a - z))
	elif z <= 3.5e+74:
		tmp = x + (((y - z) / a) * (t - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(Float64(t - x) / z) * Float64(y - a)))
	tmp = 0.0
	if (z <= -6.5e+15)
		tmp = t_1;
	elseif (z <= -2.2e-76)
		tmp = Float64(x + Float64(Float64(Float64(y - z) * t) / Float64(a - z)));
	elseif (z <= 3.5e+74)
		tmp = Float64(x + Float64(Float64(Float64(y - z) / a) * Float64(t - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - (((t - x) / z) * (y - a));
	tmp = 0.0;
	if (z <= -6.5e+15)
		tmp = t_1;
	elseif (z <= -2.2e-76)
		tmp = x + (((y - z) * t) / (a - z));
	elseif (z <= 3.5e+74)
		tmp = x + (((y - z) / a) * (t - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6.5e+15], t$95$1, If[LessEqual[z, -2.2e-76], N[(x + N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.5e+74], N[(x + N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.5e15 or 3.50000000000000014e74 < z

   1. Initial program 39.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 59.1

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. distribute-rgt-out-- N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 84.6

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

   7. Applied rewrites 84.6%

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

   if -6.5e15 < z < -2.19999999999999999e-76

   1. Initial program 88.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 88.6

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

   5. Applied rewrites 88.6%

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

   if -2.19999999999999999e-76 < z < 3.50000000000000014e74

   1. Initial program 87.3%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 77.4

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

   5. Applied rewrites 77.4%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.5 \cdot 10^{+15}:\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \mathbf{elif}\;z \leq -2.2 \cdot 10^{-76}:\\ \;\;\;\;x + \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{elif}\;z \leq 3.5 \cdot 10^{+74}:\\ \;\;\;\;x + \frac{y - z}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{z}\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(t\_1, \mathsf{fma}\left(-1, y, a\right), t\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - t\_1 \cdot \left(y - a\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- t x) z)))
   (if (<= z -4.6e+114)
     (fma t_1 (fma -1.0 y a) t)
     (if (<= z 5.5e+170)
       (fma (/ (- y z) (- a z)) (- t x) x)
       (- t (* t_1 (- y a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) / z;
	double tmp;
	if (z <= -4.6e+114) {
		tmp = fma(t_1, fma(-1.0, y, a), t);
	} else if (z <= 5.5e+170) {
		tmp = fma(((y - z) / (a - z)), (t - x), x);
	} else {
		tmp = t - (t_1 * (y - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) / z)
	tmp = 0.0
	if (z <= -4.6e+114)
		tmp = fma(t_1, fma(-1.0, y, a), t);
	elseif (z <= 5.5e+170)
		tmp = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x);
	else
		tmp = Float64(t - Float64(t_1 * Float64(y - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision]}, If[LessEqual[z, -4.6e+114], N[(t$95$1 * N[(-1.0 * y + a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 5.5e+170], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(t - N[(t$95$1 * N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.6000000000000001e114

   1. Initial program 35.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. div-sub N/A

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

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

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

      7. associate-/l* N/A

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

      8. associate-*r* N/A

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

      9. mul-1-neg N/A

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

      10. associate-/l* N/A

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

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

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

      12. distribute-rgt-out-- N/A

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

      13. lower-fma.f64 N/A

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

   5. Applied rewrites 90.2%

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

   if -4.6000000000000001e114 < z < 5.4999999999999999e170

   1. Initial program 83.9%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 89.4%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 89.5

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

      4. lift-fma.f64 N/A

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

   6. Applied rewrites 92.7%

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

   if 5.4999999999999999e170 < z

   1. Initial program 12.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 45.7

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

   4. Applied rewrites 45.7%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. distribute-rgt-out-- N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 93.7

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

   7. Applied rewrites 93.7%

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

4. Final simplification 92.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, \mathsf{fma}\left(-1, y, a\right), t\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+170}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+15} \lor \neg \left(z \leq 3.5 \cdot 10^{+74}\right):\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a} \cdot \left(t - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.5e+15) (not (<= z 3.5e+74)))
   (- t (* (/ (- t x) z) (- y a)))
   (+ x (* (/ (- y z) a) (- t x)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.5e+15) || !(z <= 3.5e+74)) {
		tmp = t - (((t - x) / z) * (y - a));
	} else {
		tmp = x + (((y - z) / a) * (t - x));
	}
	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) :: tmp
    if ((z <= (-5.5d+15)) .or. (.not. (z <= 3.5d+74))) then
        tmp = t - (((t - x) / z) * (y - a))
    else
        tmp = x + (((y - z) / a) * (t - x))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.5e+15) || !(z <= 3.5e+74)) {
		tmp = t - (((t - x) / z) * (y - a));
	} else {
		tmp = x + (((y - z) / a) * (t - x));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -5.5e+15) or not (z <= 3.5e+74):
		tmp = t - (((t - x) / z) * (y - a))
	else:
		tmp = x + (((y - z) / a) * (t - x))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.5e+15) || !(z <= 3.5e+74))
		tmp = Float64(t - Float64(Float64(Float64(t - x) / z) * Float64(y - a)));
	else
		tmp = Float64(x + Float64(Float64(Float64(y - z) / a) * Float64(t - x)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -5.5e+15) || ~((z <= 3.5e+74)))
		tmp = t - (((t - x) / z) * (y - a));
	else
		tmp = x + (((y - z) / a) * (t - x));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -5.5e+15], N[Not[LessEqual[z, 3.5e+74]], $MachinePrecision]], N[(t - N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e15 or 3.50000000000000014e74 < z

   1. Initial program 39.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 59.1

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. distribute-rgt-out-- N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 84.6

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

   7. Applied rewrites 84.6%

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

   if -5.5e15 < z < 3.50000000000000014e74

   1. Initial program 87.5%

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

   3. Taylor expanded in a around inf

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 74.9

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

   5. Applied rewrites 74.9%

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

4. Final simplification 79.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+15} \lor \neg \left(z \leq 3.5 \cdot 10^{+74}\right):\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - z}{a} \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+15} \lor \neg \left(z \leq 3.5 \cdot 10^{+74}\right):\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -5.5e+15) (not (<= z 3.5e+74)))
   (- t (* (/ (- t x) z) (- y a)))
   (fma (- y z) (/ (- t x) a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -5.5e+15) || !(z <= 3.5e+74)) {
		tmp = t - (((t - x) / z) * (y - a));
	} else {
		tmp = fma((y - z), ((t - x) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -5.5e+15) || !(z <= 3.5e+74))
		tmp = Float64(t - Float64(Float64(Float64(t - x) / z) * Float64(y - a)));
	else
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -5.5e+15], N[Not[LessEqual[z, 3.5e+74]], $MachinePrecision]], N[(t - N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -5.5e15 or 3.50000000000000014e74 < z

   1. Initial program 39.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 59.1

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

   4. Applied rewrites 59.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. lower--.f64 N/A

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

      7. div-sub N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. distribute-rgt-out-- N/A

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

      11. lower-*.f64 N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 84.6

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

   7. Applied rewrites 84.6%

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

   if -5.5e15 < z < 3.50000000000000014e74

   1. Initial program 87.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 73.8

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

   5. Applied rewrites 73.8%

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

4. Final simplification 78.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+15} \lor \neg \left(z \leq 3.5 \cdot 10^{+74}\right):\\ \;\;\;\;t - \frac{t - x}{z} \cdot \left(y - a\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 67.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+34} \lor \neg \left(a \leq 1.42 \cdot 10^{+19}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.35e+34) (not (<= a 1.42e+19)))
   (fma (- y z) (/ (- t x) a) x)
   (* (/ (- y z) (- a z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.35e+34) || !(a <= 1.42e+19)) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = ((y - z) / (a - z)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.35e+34) || !(a <= 1.42e+19))
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = Float64(Float64(Float64(y - z) / Float64(a - z)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.35e+34], N[Not[LessEqual[a, 1.42e+19]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35e34 or 1.42e19 < a

   1. Initial program 69.9%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 78.6

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

   5. Applied rewrites 78.6%

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

   if -1.35e34 < a < 1.42e19

   1. Initial program 65.4%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 70.2%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 70.3

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

      4. lift-fma.f64 N/A

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

   6. Applied rewrites 74.9%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 65.2

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

   9. Applied rewrites 65.2%

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

4. Final simplification 71.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.35 \cdot 10^{+34} \lor \neg \left(a \leq 1.42 \cdot 10^{+19}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{+46} \lor \neg \left(a \leq 2.05 \cdot 10^{+19}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -3.7e+46) (not (<= a 2.05e+19)))
   (fma (/ (- t x) a) y x)
   (* (/ (- y z) (- a z)) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -3.7e+46) || !(a <= 2.05e+19)) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = ((y - z) / (a - z)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -3.7e+46) || !(a <= 2.05e+19))
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = Float64(Float64(Float64(y - z) / Float64(a - z)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -3.7e+46], N[Not[LessEqual[a, 2.05e+19]], $MachinePrecision]], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -3.6999999999999999e46 or 2.05e19 < a

   1. Initial program 70.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 75.9

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

   5. Applied rewrites 75.9%

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

   if -3.6999999999999999e46 < a < 2.05e19

   1. Initial program 65.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 70.7%

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

      1. lift-/.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. remove-double-div 70.7

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

      4. lift-fma.f64 N/A

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

   6. Applied rewrites 75.2%

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

   7. Taylor expanded in x around 0

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

      1. associate-/l* N/A

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

      2. div-sub N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. div-sub N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 65.1

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

   9. Applied rewrites 65.1%

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

4. Final simplification 69.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.7 \cdot 10^{+46} \lor \neg \left(a \leq 2.05 \cdot 10^{+19}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{a - z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+39} \lor \neg \left(z \leq 1.2 \cdot 10^{+140}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.1e+39) (not (<= z 1.2e+140)))
   (* (- t) (/ z (- a z)))
   (fma (/ (- t x) a) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.1e+39) || !(z <= 1.2e+140)) {
		tmp = -t * (z / (a - z));
	} else {
		tmp = fma(((t - x) / a), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.1e+39) || !(z <= 1.2e+140))
		tmp = Float64(Float64(-t) * Float64(z / Float64(a - z)));
	else
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.1e+39], N[Not[LessEqual[z, 1.2e+140]], $MachinePrecision]], N[((-t) * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1000000000000001e39 or 1.2e140 < z

   1. Initial program 34.7%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 28.0

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 62.9%

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

   if -1.1000000000000001e39 < z < 1.2e140

   1. Initial program 84.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 67.5

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

   5. Applied rewrites 67.5%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+39} \lor \neg \left(z \leq 1.2 \cdot 10^{+140}\right):\\ \;\;\;\;\left(-t\right) \cdot \frac{z}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+78} \lor \neg \left(z \leq 1.75 \cdot 10^{+140}\right):\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.22e+78) (not (<= z 1.75e+140))) t (fma (- y z) (/ t a) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.22e+78) || !(z <= 1.75e+140)) {
		tmp = t;
	} else {
		tmp = fma((y - z), (t / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.22e+78) || !(z <= 1.75e+140))
		tmp = t;
	else
		tmp = fma(Float64(y - z), Float64(t / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.22e+78], N[Not[LessEqual[z, 1.75e+140]], $MachinePrecision]], t, N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.22000000000000006e78 or 1.74999999999999995e140 < z

   1. Initial program 30.2%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 53.5%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 25.4

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

   7. Applied rewrites 25.4%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 64.2%

       \[\leadsto t \]

   if -1.22000000000000006e78 < z < 1.74999999999999995e140

   1. Initial program 84.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 68.6

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

   5. Applied rewrites 68.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 58.5%

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

4. Final simplification 60.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.22 \cdot 10^{+78} \lor \neg \left(z \leq 1.75 \cdot 10^{+140}\right):\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+39}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.9e+39) t (if (<= z 1.2e+140) (fma (/ (- t x) a) y x) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.9e+39) {
		tmp = t;
	} else if (z <= 1.2e+140) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.9e+39)
		tmp = t;
	elseif (z <= 1.2e+140)
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.9e+39], t, If[LessEqual[z, 1.2e+140], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8999999999999999e39 or 1.2e140 < z

   1. Initial program 34.7%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 56.1%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 28.0

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

   7. Applied rewrites 28.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 61.6%

       \[\leadsto t \]

   if -1.8999999999999999e39 < z < 1.2e140

   1. Initial program 84.6%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 67.5

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

   5. Applied rewrites 67.5%

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

4. Final simplification 65.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+39}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 1.2 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+39} \lor \neg \left(z \leq 1.9 \cdot 10^{+111}\right):\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.9e+39) (not (<= z 1.9e+111))) t (- x (* x (/ y a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.9e+39) || !(z <= 1.9e+111)) {
		tmp = t;
	} else {
		tmp = x - (x * (y / a));
	}
	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) :: tmp
    if ((z <= (-1.9d+39)) .or. (.not. (z <= 1.9d+111))) then
        tmp = t
    else
        tmp = x - (x * (y / a))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.9e+39) || !(z <= 1.9e+111)) {
		tmp = t;
	} else {
		tmp = x - (x * (y / a));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.9e+39) or not (z <= 1.9e+111):
		tmp = t
	else:
		tmp = x - (x * (y / a))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.9e+39) || !(z <= 1.9e+111))
		tmp = t;
	else
		tmp = Float64(x - Float64(x * Float64(y / a)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.9e+39) || ~((z <= 1.9e+111)))
		tmp = t;
	else
		tmp = x - (x * (y / a));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.9e+39], N[Not[LessEqual[z, 1.9e+111]], $MachinePrecision]], t, N[(x - N[(x * N[(y / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.8999999999999999e39 or 1.89999999999999988e111 < z

   1. Initial program 34.8%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 57.5%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 26.5

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

   7. Applied rewrites 26.5%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 58.2%

       \[\leadsto t \]

   if -1.8999999999999999e39 < z < 1.89999999999999988e111

   1. Initial program 86.7%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 71.6

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

   5. Applied rewrites 71.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 49.9%

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

   9. Taylor expanded in y around inf

      \[\leadsto x - x \cdot \frac{y}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 49.9%

       \[\leadsto x - x \cdot \frac{y}{a} \]
3. Recombined 2 regimes into one program.

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.9 \cdot 10^{+39} \lor \neg \left(z \leq 1.9 \cdot 10^{+111}\right):\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x - x \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+34} \lor \neg \left(a \leq 1.9 \cdot 10^{+94}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -1.4e+34) (not (<= a 1.9e+94))) (fma x (/ z a) x) t))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -1.4e+34) || !(a <= 1.9e+94)) {
		tmp = fma(x, (z / a), x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -1.4e+34) || !(a <= 1.9e+94))
		tmp = fma(x, Float64(z / a), x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -1.4e+34], N[Not[LessEqual[a, 1.9e+94]], $MachinePrecision]], N[(x * N[(z / a), $MachinePrecision] + x), $MachinePrecision], t]

Derivation

1. Split input into 2 regimes

2. if a < -1.40000000000000004e34 or 1.8999999999999998e94 < a

   1. Initial program 72.2%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 80.6

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

   5. Applied rewrites 80.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 63.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 58.2%

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

   if -1.40000000000000004e34 < a < 1.8999999999999998e94

   1. Initial program 64.4%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 70.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 24.0

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

   7. Applied rewrites 24.0%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 39.1%

       \[\leadsto t \]
3. Recombined 2 regimes into one program.

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.4 \cdot 10^{+34} \lor \neg \left(a \leq 1.9 \cdot 10^{+94}\right):\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 31.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-7} \lor \neg \left(a \leq 5.2 \cdot 10^{+109}\right):\\ \;\;\;\;\frac{x \cdot a}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -7e-7) (not (<= a 5.2e+109))) (/ (* x a) a) t))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7e-7) || !(a <= 5.2e+109)) {
		tmp = (x * a) / a;
	} else {
		tmp = t;
	}
	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) :: tmp
    if ((a <= (-7d-7)) .or. (.not. (a <= 5.2d+109))) then
        tmp = (x * a) / a
    else
        tmp = t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -7e-7) || !(a <= 5.2e+109)) {
		tmp = (x * a) / a;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (a <= -7e-7) or not (a <= 5.2e+109):
		tmp = (x * a) / a
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -7e-7) || !(a <= 5.2e+109))
		tmp = Float64(Float64(x * a) / a);
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((a <= -7e-7) || ~((a <= 5.2e+109)))
		tmp = (x * a) / a;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -7e-7], N[Not[LessEqual[a, 5.2e+109]], $MachinePrecision]], N[(N[(x * a), $MachinePrecision] / a), $MachinePrecision], t]

Derivation

1. Split input into 2 regimes

2. if a < -6.99999999999999968e-7 or 5.1999999999999997e109 < a

   1. Initial program 70.7%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 77.5

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

   5. Applied rewrites 77.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 56.5%

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

   9. Taylor expanded in a around inf

      \[\leadsto \frac{a \cdot x}{a} \]
   10. Step-by-step derivation

   11. Applied rewrites 41.1%

       \[\leadsto \frac{x \cdot a}{a} \]

   if -6.99999999999999968e-7 < a < 5.1999999999999997e109

   1. Initial program 65.1%

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

      1. lift-+.f64 N/A

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

      2. flip3-+ N/A

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

      3. clear-num N/A

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

      4. lower-/.f64 N/A

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

      5. clear-num N/A

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

      6. flip3-+ N/A

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

   4. Applied rewrites 69.7%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower--.f64 23.1

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

   7. Applied rewrites 23.1%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 39.7%

       \[\leadsto t \]
3. Recombined 2 regimes into one program.

4. Final simplification 40.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7 \cdot 10^{-7} \lor \neg \left(a \leq 5.2 \cdot 10^{+109}\right):\\ \;\;\;\;\frac{x \cdot a}{a}\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 25.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ t \end{array} \]

FPCore

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

C

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

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 = t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 67.4%

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

   1. lift-+.f64 N/A

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

   2. flip3-+ N/A

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

   3. clear-num N/A

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

   4. lower-/.f64 N/A

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

   5. clear-num N/A

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

   6. flip3-+ N/A

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

4. Applied rewrites 77.7%

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

5. Taylor expanded in y around 0

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 N/A

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

   4. lower-/.f64 N/A

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

   5. *-commutative N/A

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

   6. lower-*.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower--.f64 38.7

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

7. Applied rewrites 38.7%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 28.3%

    \[\leadsto t \]
11. Add Preprocessing

Developer Target 1: 83.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ y z) (- t x)))))
   (if (< z -1.2536131056095036e+188)
     t_1
     (if (< z 4.446702369113811e+64)
       (+ x (/ (- y z) (/ (- a z) (- t x))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} 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) :: tmp
    t_1 = t - ((y / z) * (t - x))
    if (z < (-1.2536131056095036d+188)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x + ((y - z) / ((a - z) / (t - x)))
    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 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - ((y / z) * (t - x))
	tmp = 0
	if z < -1.2536131056095036e+188:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x + ((y - z) / ((a - z) / (t - x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(y / z) * Float64(t - x)))
	tmp = 0.0
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x + Float64(Float64(y - z) / Float64(Float64(a - z) / Float64(t - x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - ((y / z) * (t - x));
	tmp = 0.0;
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(y / z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.2536131056095036e+188], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x + N[(N[(y - z), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024324 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -125361310560950360000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- t (* (/ y z) (- t x))) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x))))))

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