Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 80.6% → 90.9%

Time: 10.7s

Alternatives: 20

Speedup: 0.3×

Specification

Math

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

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(y - z) * Float64(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[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 80.6% accurate, 1.0× speedup

Math

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

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(y - z) * Float64(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[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{y - z}{\frac{z - a}{x - t}} + x\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* (/ (- t x) (- a z)) (- y z)) x)))
   (if (<= t_1 -5e-180)
     (+ (/ (- y z) (/ (- z a) (- x t))) x)
     (if (<= t_1 2e-290)
       (fma (- y a) (/ (- x t) z) t)
       (fma (/ (- x t) (- z a)) (- y z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (((t - x) / (a - z)) * (y - z)) + x;
	double tmp;
	if (t_1 <= -5e-180) {
		tmp = ((y - z) / ((z - a) / (x - t))) + x;
	} else if (t_1 <= 2e-290) {
		tmp = fma((y - a), ((x - t) / z), t);
	} else {
		tmp = fma(((x - t) / (z - a)), (y - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(Float64(t - x) / Float64(a - z)) * Float64(y - z)) + x)
	tmp = 0.0
	if (t_1 <= -5e-180)
		tmp = Float64(Float64(Float64(y - z) / Float64(Float64(z - a) / Float64(x - t))) + x);
	elseif (t_1 <= 2e-290)
		tmp = fma(Float64(y - a), Float64(Float64(x - t) / z), t);
	else
		tmp = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-180], N[(N[(N[(y - z), $MachinePrecision] / N[(N[(z - a), $MachinePrecision] / N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e-290], N[(N[(y - a), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -5.0000000000000001e-180

   1. Initial program 96.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. clear-num N/A

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

         \[\leadsto x + \frac{y - z}{\frac{\color{blue}{0 - \left(a - z\right)}}{\mathsf{neg}\left(\left(t - x\right)\right)}} \]

      9. lift--.f64 N/A

         \[\leadsto x + \frac{y - z}{\frac{0 - \color{blue}{\left(a - z\right)}}{\mathsf{neg}\left(\left(t - x\right)\right)}} \]

      10. sub-neg N/A

          \[\leadsto x + \frac{y - z}{\frac{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(z\right)\right)\right)}}{\mathsf{neg}\left(\left(t - x\right)\right)}} \]

      11. +-commutative N/A

          \[\leadsto x + \frac{y - z}{\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + a\right)}}{\mathsf{neg}\left(\left(t - x\right)\right)}} \]

      12. associate--r+ N/A

          \[\leadsto x + \frac{y - z}{\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - a}}{\mathsf{neg}\left(\left(t - x\right)\right)}} \]

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

          \[\leadsto x + \frac{y - z}{\frac{z - a}{\color{blue}{0 - \left(t - x\right)}}} \]

      17. lift--.f64 N/A

          \[\leadsto x + \frac{y - z}{\frac{z - a}{0 - \color{blue}{\left(t - x\right)}}} \]

      18. sub-neg N/A

          \[\leadsto x + \frac{y - z}{\frac{z - a}{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)}}} \]

      19. +-commutative N/A

          \[\leadsto x + \frac{y - z}{\frac{z - a}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + t\right)}}} \]

      20. associate--r+ N/A

          \[\leadsto x + \frac{y - z}{\frac{z - a}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - t}}} \]

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 97.2

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

   4. Applied rewrites 97.2%

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

   if -5.0000000000000001e-180 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 2.0000000000000001e-290

   1. Initial program 8.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 91.1%

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

   10. Applied rewrites 94.7%

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

   if 2.0000000000000001e-290 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 89.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 89.4

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - x\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(t - x\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + t\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - t}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{0 - \left(a - z\right)}}, y - z, x\right) \]

      18. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(a - z\right)}}, y - z, x\right) \]

      19. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(z\right)\right)\right)}}, y - z, x\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + a\right)}}, y - z, x\right) \]

      21. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - a}}, y - z, x\right) \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 89.4

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

   4. Applied rewrites 89.4%

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

4. Final simplification 93.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{y - z}{\frac{z - a}{x - t}} + x\\ \mathbf{elif}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-180}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* (/ (- t x) (- a z)) (- y z)) x)))
   (if (<= t_1 -5e-180)
     t_1
     (if (<= t_1 2e-290)
       (fma (- y a) (/ (- x t) z) t)
       (fma (/ (- x t) (- z a)) (- y z) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (((t - x) / (a - z)) * (y - z)) + x;
	double tmp;
	if (t_1 <= -5e-180) {
		tmp = t_1;
	} else if (t_1 <= 2e-290) {
		tmp = fma((y - a), ((x - t) / z), t);
	} else {
		tmp = fma(((x - t) / (z - a)), (y - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(Float64(t - x) / Float64(a - z)) * Float64(y - z)) + x)
	tmp = 0.0
	if (t_1 <= -5e-180)
		tmp = t_1;
	elseif (t_1 <= 2e-290)
		tmp = fma(Float64(y - a), Float64(Float64(x - t) / z), t);
	else
		tmp = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-180], t$95$1, If[LessEqual[t$95$1, 2e-290], N[(N[(y - a), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -5.0000000000000001e-180

   1. Initial program 96.8%

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

   if -5.0000000000000001e-180 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 2.0000000000000001e-290

   1. Initial program 8.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 91.1%

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

   10. Applied rewrites 94.7%

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

   if 2.0000000000000001e-290 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 89.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 89.4

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - x\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(t - x\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + t\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - t}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{0 - \left(a - z\right)}}, y - z, x\right) \]

      18. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(a - z\right)}}, y - z, x\right) \]

      19. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(z\right)\right)\right)}}, y - z, x\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + a\right)}}, y - z, x\right) \]

      21. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - a}}, y - z, x\right) \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 89.4

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

   4. Applied rewrites 89.4%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \mathbf{elif}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ t_2 := \frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-180}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) (- z a)) (- y z) x))
        (t_2 (+ (* (/ (- t x) (- a z)) (- y z)) x)))
   (if (<= t_2 -5e-180)
     t_1
     (if (<= t_2 2e-290) (fma (- y a) (/ (- x t) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / (z - a)), (y - z), x);
	double t_2 = (((t - x) / (a - z)) * (y - z)) + x;
	double tmp;
	if (t_2 <= -5e-180) {
		tmp = t_1;
	} else if (t_2 <= 2e-290) {
		tmp = fma((y - a), ((x - t) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x)
	t_2 = Float64(Float64(Float64(Float64(t - x) / Float64(a - z)) * Float64(y - z)) + x)
	tmp = 0.0
	if (t_2 <= -5e-180)
		tmp = t_1;
	elseif (t_2 <= 2e-290)
		tmp = fma(Float64(y - a), Float64(Float64(x - t) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-180], t$95$1, If[LessEqual[t$95$2, 2e-290], N[(N[(y - a), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -5.0000000000000001e-180 or 2.0000000000000001e-290 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 93.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.1

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

         \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{0 - \left(t - x\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      10. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(t - x\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      11. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      12. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{0 - \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + t\right)}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      13. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{\color{blue}{\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - t}}{\mathsf{neg}\left(\left(a - z\right)\right)}, y - z, x\right) \]

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{0 - \left(a - z\right)}}, y - z, x\right) \]

      18. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(a - z\right)}}, y - z, x\right) \]

      19. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(z\right)\right)\right)}}, y - z, x\right) \]

      20. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(z\right)\right) + a\right)}}, y - z, x\right) \]

      21. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{x - t}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(z\right)\right)\right) - a}}, y - z, x\right) \]

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 93.1

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

   4. Applied rewrites 93.1%

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

   if -5.0000000000000001e-180 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 2.0000000000000001e-290

   1. Initial program 8.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 91.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 91.1%

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

   10. Applied rewrites 94.7%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq -5 \cdot 10^{-180}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \mathbf{elif}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq 2 \cdot 10^{-290}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 78.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+32}:\\ \;\;\;\;\frac{y}{a - z} \cdot \left(t - x\right) + x\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.18e+52)
   (fma (- y a) (/ (- x t) z) t)
   (if (<= z 1.7e+32)
     (+ (* (/ y (- a z)) (- t x)) x)
     (if (<= z 3.1e+146)
       (* (/ t (- a z)) (- y z))
       (fma (- x t) (/ (- y a) z) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.18e+52) {
		tmp = fma((y - a), ((x - t) / z), t);
	} else if (z <= 1.7e+32) {
		tmp = ((y / (a - z)) * (t - x)) + x;
	} else if (z <= 3.1e+146) {
		tmp = (t / (a - z)) * (y - z);
	} else {
		tmp = fma((x - t), ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.18e+52)
		tmp = fma(Float64(y - a), Float64(Float64(x - t) / z), t);
	elseif (z <= 1.7e+32)
		tmp = Float64(Float64(Float64(y / Float64(a - z)) * Float64(t - x)) + x);
	elseif (z <= 3.1e+146)
		tmp = Float64(Float64(t / Float64(a - z)) * Float64(y - z));
	else
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.18e+52], N[(N[(y - a), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 1.7e+32], N[(N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3.1e+146], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.17999999999999997e52

   1. Initial program 64.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 82.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 83.3%

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

   10. Applied rewrites 84.0%

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

   if -1.17999999999999997e52 < z < 1.69999999999999989e32

   1. Initial program 94.2%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if 1.69999999999999989e32 < z < 3.1000000000000002e146

   1. Initial program 87.8%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if 3.1000000000000002e146 < z

   1. Initial program 51.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 87.2%

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

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.7 \cdot 10^{+32}:\\ \;\;\;\;\frac{y}{a - z} \cdot \left(t - x\right) + x\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 73.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-25)
   (fma (- y a) (/ (- x t) z) t)
   (if (<= z 1.15e+32)
     (fma (/ (- y z) a) (- t x) x)
     (if (<= z 3.1e+146)
       (* (/ t (- a z)) (- y z))
       (fma (- x t) (/ (- y a) z) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-25) {
		tmp = fma((y - a), ((x - t) / z), t);
	} else if (z <= 1.15e+32) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else if (z <= 3.1e+146) {
		tmp = (t / (a - z)) * (y - z);
	} else {
		tmp = fma((x - t), ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-25)
		tmp = fma(Float64(y - a), Float64(Float64(x - t) / z), t);
	elseif (z <= 1.15e+32)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	elseif (z <= 3.1e+146)
		tmp = Float64(Float64(t / Float64(a - z)) * Float64(y - z));
	else
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-25], N[(N[(y - a), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 1.15e+32], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3.1e+146], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.00000000000000004e-25

   1. Initial program 69.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 78.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 81.1%

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

   10. Applied rewrites 81.8%

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

   if -1.00000000000000004e-25 < z < 1.15e32

   1. Initial program 94.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if 1.15e32 < z < 3.1000000000000002e146

   1. Initial program 87.8%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if 3.1000000000000002e146 < z

   1. Initial program 51.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 87.2%

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

4. Final simplification 81.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-25}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.5e-32)
   (fma (- y a) (/ (- x t) z) t)
   (if (<= z 1.06e+32)
     (fma (/ y a) (- t x) x)
     (if (<= z 3.1e+146)
       (* (/ t (- a z)) (- y z))
       (fma (- x t) (/ (- y a) z) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.5e-32) {
		tmp = fma((y - a), ((x - t) / z), t);
	} else if (z <= 1.06e+32) {
		tmp = fma((y / a), (t - x), x);
	} else if (z <= 3.1e+146) {
		tmp = (t / (a - z)) * (y - z);
	} else {
		tmp = fma((x - t), ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.5e-32)
		tmp = fma(Float64(y - a), Float64(Float64(x - t) / z), t);
	elseif (z <= 1.06e+32)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	elseif (z <= 3.1e+146)
		tmp = Float64(Float64(t / Float64(a - z)) * Float64(y - z));
	else
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.5e-32], N[(N[(y - a), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 1.06e+32], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3.1e+146], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.5e-32

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 80.3%

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

   10. Applied rewrites 81.0%

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

   if -1.5e-32 < z < 1.0600000000000001e32

   1. Initial program 94.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.5%

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

   if 1.0600000000000001e32 < z < 3.1000000000000002e146

   1. Initial program 87.8%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if 3.1000000000000002e146 < z

   1. Initial program 51.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 87.2%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.5 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(y - a, \frac{x - t}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 71.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ (- y a) z) t)))
   (if (<= z -1.4e-32)
     t_1
     (if (<= z 1.06e+32)
       (fma (/ y a) (- t x) x)
       (if (<= z 3.1e+146) (* (/ t (- a z)) (- y z)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), ((y - a) / z), t);
	double tmp;
	if (z <= -1.4e-32) {
		tmp = t_1;
	} else if (z <= 1.06e+32) {
		tmp = fma((y / a), (t - x), x);
	} else if (z <= 3.1e+146) {
		tmp = (t / (a - z)) * (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(Float64(y - a) / z), t)
	tmp = 0.0
	if (z <= -1.4e-32)
		tmp = t_1;
	elseif (z <= 1.06e+32)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	elseif (z <= 3.1e+146)
		tmp = Float64(Float64(t / Float64(a - z)) * Float64(y - z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.4e-32], t$95$1, If[LessEqual[z, 1.06e+32], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3.1e+146], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3999999999999999e-32 or 3.1000000000000002e146 < z

   1. Initial program 64.1%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 80.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 82.6%

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

   if -1.3999999999999999e-32 < z < 1.0600000000000001e32

   1. Initial program 94.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.5%

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

   if 1.0600000000000001e32 < z < 3.1000000000000002e146

   1. Initial program 87.8%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+146}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 68.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-31)
   (fma (/ (- x t) z) y t)
   (if (<= z 1.06e+32)
     (fma (/ y a) (- t x) x)
     (if (<= z 5.5e+146) (* (/ t (- a z)) (- y z)) (fma (- x t) (/ y z) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-31) {
		tmp = fma(((x - t) / z), y, t);
	} else if (z <= 1.06e+32) {
		tmp = fma((y / a), (t - x), x);
	} else if (z <= 5.5e+146) {
		tmp = (t / (a - z)) * (y - z);
	} else {
		tmp = fma((x - t), (y / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-31)
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	elseif (z <= 1.06e+32)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	elseif (z <= 5.5e+146)
		tmp = Float64(Float64(t / Float64(a - z)) * Float64(y - z));
	else
		tmp = fma(Float64(x - t), Float64(y / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-31], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], If[LessEqual[z, 1.06e+32], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 5.5e+146], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1e-31

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.1%

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

   if -1e-31 < z < 1.0600000000000001e32

   1. Initial program 94.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.5%

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

   if 1.0600000000000001e32 < z < 5.5000000000000004e146

   1. Initial program 87.8%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   if 5.5000000000000004e146 < z

   1. Initial program 51.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 86.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 87.2%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 79.5%

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

4. Final simplification 76.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{elif}\;z \leq 1.06 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;z \leq 5.5 \cdot 10^{+146}:\\ \;\;\;\;\frac{t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 41.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(y - a\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t) (/ y z) t)))
   (if (<= z -4e-12)
     t_1
     (if (<= z -4.3e-35)
       (/ (* (- y a) x) z)
       (if (<= z 1.1e-40) (* (/ y (- a z)) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-t, (y / z), t);
	double tmp;
	if (z <= -4e-12) {
		tmp = t_1;
	} else if (z <= -4.3e-35) {
		tmp = ((y - a) * x) / z;
	} else if (z <= 1.1e-40) {
		tmp = (y / (a - z)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -4e-12)
		tmp = t_1;
	elseif (z <= -4.3e-35)
		tmp = Float64(Float64(Float64(y - a) * x) / z);
	elseif (z <= 1.1e-40)
		tmp = Float64(Float64(y / Float64(a - z)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -4e-12], t$95$1, If[LessEqual[z, -4.3e-35], N[(N[(N[(y - a), $MachinePrecision] * x), $MachinePrecision] / z), $MachinePrecision], If[LessEqual[z, 1.1e-40], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.99999999999999992e-12 or 1.10000000000000004e-40 < z

   1. Initial program 70.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 73.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 68.2%

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

   12. Taylor expanded in t around inf

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

   14. Applied rewrites 58.0%

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

   if -3.99999999999999992e-12 < z < -4.3000000000000002e-35

   1. Initial program 75.7%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 62.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 74.9%

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

   9. Taylor expanded in t around 0

      \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]
   10. Step-by-step derivation

   11. Applied rewrites 68.0%

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

   if -4.3000000000000002e-35 < z < 1.10000000000000004e-40

   1. Initial program 94.3%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 66.3

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 42.0%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{elif}\;z \leq -4.3 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(y - a\right) \cdot x}{z}\\ \mathbf{elif}\;z \leq 1.1 \cdot 10^{-40}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 30.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y}{a - z} \cdot t\\ \mathbf{if}\;y \leq -6.6 \cdot 10^{+147}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+73}:\\ \;\;\;\;\frac{y - a}{z} \cdot x\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+16}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ y (- a z)) t)))
   (if (<= y -6.6e+147)
     t_1
     (if (<= y -2.5e+73)
       (* (/ (- y a) z) x)
       (if (<= y 6.9e+16) (+ (- t x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y / (a - z)) * t;
	double tmp;
	if (y <= -6.6e+147) {
		tmp = t_1;
	} else if (y <= -2.5e+73) {
		tmp = ((y - a) / z) * x;
	} else if (y <= 6.9e+16) {
		tmp = (t - x) + 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 = (y / (a - z)) * t
    if (y <= (-6.6d+147)) then
        tmp = t_1
    else if (y <= (-2.5d+73)) then
        tmp = ((y - a) / z) * x
    else if (y <= 6.9d+16) then
        tmp = (t - x) + 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 = (y / (a - z)) * t;
	double tmp;
	if (y <= -6.6e+147) {
		tmp = t_1;
	} else if (y <= -2.5e+73) {
		tmp = ((y - a) / z) * x;
	} else if (y <= 6.9e+16) {
		tmp = (t - x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y / (a - z)) * t
	tmp = 0
	if y <= -6.6e+147:
		tmp = t_1
	elif y <= -2.5e+73:
		tmp = ((y - a) / z) * x
	elif y <= 6.9e+16:
		tmp = (t - x) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y / Float64(a - z)) * t)
	tmp = 0.0
	if (y <= -6.6e+147)
		tmp = t_1;
	elseif (y <= -2.5e+73)
		tmp = Float64(Float64(Float64(y - a) / z) * x);
	elseif (y <= 6.9e+16)
		tmp = Float64(Float64(t - x) + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y / (a - z)) * t;
	tmp = 0.0;
	if (y <= -6.6e+147)
		tmp = t_1;
	elseif (y <= -2.5e+73)
		tmp = ((y - a) / z) * x;
	elseif (y <= 6.9e+16)
		tmp = (t - x) + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[y, -6.6e+147], t$95$1, If[LessEqual[y, -2.5e+73], N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 6.9e+16], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -6.60000000000000049e147 or 6.9e16 < y

   1. Initial program 86.4%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 77.1

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

   5. Applied rewrites 77.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 52.0%

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

   if -6.60000000000000049e147 < y < -2.49999999999999988e73

   1. Initial program 80.5%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 47.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 55.1%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 55.1%

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

   12. Taylor expanded in t around 0

       \[\leadsto \frac{x \cdot \left(y - a\right)}{z} \]
   13. Step-by-step derivation

   14. Applied rewrites 48.1%

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

   if -2.49999999999999988e73 < y < 6.9e16

   1. Initial program 76.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 37.1

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

   5. Applied rewrites 37.1%

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

4. Final simplification 43.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -6.6 \cdot 10^{+147}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \mathbf{elif}\;y \leq -2.5 \cdot 10^{+73}:\\ \;\;\;\;\frac{y - a}{z} \cdot x\\ \mathbf{elif}\;y \leq 6.9 \cdot 10^{+16}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 69.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{+32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-31)
   (fma (/ (- x t) z) y t)
   (if (<= z 1.15e+32) (fma (/ y a) (- t x) x) (fma (- x t) (/ y z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-31) {
		tmp = fma(((x - t) / z), y, t);
	} else if (z <= 1.15e+32) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = fma((x - t), (y / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-31)
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	elseif (z <= 1.15e+32)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = fma(Float64(x - t), Float64(y / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-31], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], If[LessEqual[z, 1.15e+32], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1e-31

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.1%

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

   if -1e-31 < z < 1.15e32

   1. Initial program 94.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 76.5%

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

   if 1.15e32 < z

   1. Initial program 65.6%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 76.8%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 78.4%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 73.4%

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

Alternative 12: 55.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e-32)
   (fma (/ (- x t) z) y t)
   (if (<= z 5.7e-117) (* (/ y a) (- t x)) (fma (- x t) (/ y z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e-32) {
		tmp = fma(((x - t) / z), y, t);
	} else if (z <= 5.7e-117) {
		tmp = (y / a) * (t - x);
	} else {
		tmp = fma((x - t), (y / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e-32)
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	elseif (z <= 5.7e-117)
		tmp = Float64(Float64(y / a) * Float64(t - x));
	else
		tmp = fma(Float64(x - t), Float64(y / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e-32], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision], If[LessEqual[z, 5.7e-117], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.3499999999999999e-32

   1. Initial program 70.4%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 71.1%

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

   if -1.3499999999999999e-32 < z < 5.6999999999999999e-117

   1. Initial program 95.6%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 58.7%

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

   if 5.6999999999999999e-117 < z

   1. Initial program 73.7%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 65.6%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 67.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 63.2%

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

4. Final simplification 64.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 55.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{-32}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x t) (/ y z) t)))
   (if (<= z -1.35e-32) t_1 (if (<= z 5.7e-117) (* (/ y a) (- t x)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - t), (y / z), t);
	double tmp;
	if (z <= -1.35e-32) {
		tmp = t_1;
	} else if (z <= 5.7e-117) {
		tmp = (y / a) * (t - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -1.35e-32)
		tmp = t_1;
	elseif (z <= 5.7e-117)
		tmp = Float64(Float64(y / a) * Float64(t - x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.35e-32], t$95$1, If[LessEqual[z, 5.7e-117], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.3499999999999999e-32 or 5.6999999999999999e-117 < z

   1. Initial program 72.2%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 71.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 73.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 66.5%

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

   if -1.3499999999999999e-32 < z < 5.6999999999999999e-117

   1. Initial program 95.6%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 58.7%

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

4. Final simplification 63.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{elif}\;z \leq 5.7 \cdot 10^{-117}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 45.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t) (/ y z) t)))
   (if (<= z -6.4e-9) t_1 (if (<= z 5.8e-34) (* (/ y a) (- t x)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-t, (y / z), t);
	double tmp;
	if (z <= -6.4e-9) {
		tmp = t_1;
	} else if (z <= 5.8e-34) {
		tmp = (y / a) * (t - x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -6.4e-9)
		tmp = t_1;
	elseif (z <= 5.8e-34)
		tmp = Float64(Float64(y / a) * Float64(t - x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -6.4e-9], t$95$1, If[LessEqual[z, 5.8e-34], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.40000000000000023e-9 or 5.8000000000000004e-34 < z

   1. Initial program 70.2%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 68.4%

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

   12. Taylor expanded in t around inf

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

   14. Applied rewrites 58.1%

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

   if -6.40000000000000023e-9 < z < 5.8000000000000004e-34

   1. Initial program 93.1%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 66.8

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

   5. Applied rewrites 66.8%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 53.8%

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

4. Final simplification 56.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.4 \cdot 10^{-9}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{-34}:\\ \;\;\;\;\frac{y}{a} \cdot \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 44.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -5.6 \cdot 10^{-9}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{-35}:\\ \;\;\;\;\frac{\left(t - x\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t) (/ y z) t)))
   (if (<= z -5.6e-9) t_1 (if (<= z 7.2e-35) (/ (* (- t x) y) a) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-t, (y / z), t);
	double tmp;
	if (z <= -5.6e-9) {
		tmp = t_1;
	} else if (z <= 7.2e-35) {
		tmp = ((t - x) * y) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -5.6e-9)
		tmp = t_1;
	elseif (z <= 7.2e-35)
		tmp = Float64(Float64(Float64(t - x) * y) / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -5.6e-9], t$95$1, If[LessEqual[z, 7.2e-35], N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.59999999999999969e-9 or 7.20000000000000038e-35 < z

   1. Initial program 70.2%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 73.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 75.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 68.4%

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

   12. Taylor expanded in t around inf

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

   14. Applied rewrites 58.1%

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

   if -5.59999999999999969e-9 < z < 7.20000000000000038e-35

   1. Initial program 93.1%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 79.3

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

   5. Applied rewrites 79.3%

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

   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 47.9%

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

Alternative 16: 41.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -7.8 \cdot 10^{-26}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.4 \cdot 10^{-10}:\\ \;\;\;\;\frac{y - z}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t) (/ y z) t)))
   (if (<= z -7.8e-26) t_1 (if (<= z 3.4e-10) (* (/ (- y z) a) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-t, (y / z), t);
	double tmp;
	if (z <= -7.8e-26) {
		tmp = t_1;
	} else if (z <= 3.4e-10) {
		tmp = ((y - z) / a) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -7.8e-26)
		tmp = t_1;
	elseif (z <= 3.4e-10)
		tmp = Float64(Float64(Float64(y - z) / a) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -7.8e-26], t$95$1, If[LessEqual[z, 3.4e-10], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.79999999999999973e-26 or 3.40000000000000015e-10 < z

   1. Initial program 70.0%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 74.5%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 77.3%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 70.2%

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

   12. Taylor expanded in t around inf

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

   14. Applied rewrites 58.0%

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

   if -7.79999999999999973e-26 < z < 3.40000000000000015e-10

   1. Initial program 93.8%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 79.8

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

   5. Applied rewrites 79.8%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 42.6%

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

Alternative 17: 32.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) + x\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+45}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- t x) x)))
   (if (<= z -1.4e+52) t_1 (if (<= z 7e+45) (* (/ y (- a z)) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) + x;
	double tmp;
	if (z <= -1.4e+52) {
		tmp = t_1;
	} else if (z <= 7e+45) {
		tmp = (y / (a - z)) * t;
	} 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 - x) + x
    if (z <= (-1.4d+52)) then
        tmp = t_1
    else if (z <= 7d+45) then
        tmp = (y / (a - z)) * t
    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 - x) + x;
	double tmp;
	if (z <= -1.4e+52) {
		tmp = t_1;
	} else if (z <= 7e+45) {
		tmp = (y / (a - z)) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (t - x) + x
	tmp = 0
	if z <= -1.4e+52:
		tmp = t_1
	elif z <= 7e+45:
		tmp = (y / (a - z)) * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) + x)
	tmp = 0.0
	if (z <= -1.4e+52)
		tmp = t_1;
	elseif (z <= 7e+45)
		tmp = Float64(Float64(y / Float64(a - z)) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (t - x) + x;
	tmp = 0.0;
	if (z <= -1.4e+52)
		tmp = t_1;
	elseif (z <= 7e+45)
		tmp = (y / (a - z)) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.4e+52], t$95$1, If[LessEqual[z, 7e+45], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4e52 or 7.00000000000000046e45 < z

   1. Initial program 65.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 42.9

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

   5. Applied rewrites 42.9%

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

   if -1.4e52 < z < 7.00000000000000046e45

   1. Initial program 93.7%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 62.1

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

   5. Applied rewrites 62.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 39.5%

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

4. Final simplification 41.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+52}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{elif}\;z \leq 7 \cdot 10^{+45}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 19.7% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{+111}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a 3.6e+111) (+ (- t x) x) (* (/ (- z) a) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= 3.6e+111) {
		tmp = (t - x) + x;
	} else {
		tmp = (-z / a) * 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 <= 3.6d+111) then
        tmp = (t - x) + x
    else
        tmp = (-z / a) * 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 <= 3.6e+111) {
		tmp = (t - x) + x;
	} else {
		tmp = (-z / a) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= 3.6e+111:
		tmp = (t - x) + x
	else:
		tmp = (-z / a) * t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= 3.6e+111)
		tmp = Float64(Float64(t - x) + x);
	else
		tmp = Float64(Float64(Float64(-z) / a) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= 3.6e+111)
		tmp = (t - x) + x;
	else
		tmp = (-z / a) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, 3.6e+111], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[((-z) / a), $MachinePrecision] * t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < 3.6000000000000002e111

   1. Initial program 78.9%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 28.9

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

   5. Applied rewrites 28.9%

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

   if 3.6000000000000002e111 < a

   1. Initial program 87.6%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 87.3

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

   5. Applied rewrites 87.3%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 17.9%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 18.6%

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

4. Final simplification 27.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq 3.6 \cdot 10^{+111}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\frac{-z}{a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 19.4% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ \left(t - x\right) + x \end{array} \]

FPCore

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

C

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

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 - x) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 25.2

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

5. Applied rewrites 25.2%

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

6. Final simplification 25.2%

   \[\leadsto \left(t - x\right) + x \]
7. Add Preprocessing

Alternative 20: 2.8% accurate, 4.8× speedup

Math

\[\begin{array}{l} \\ \left(-x\right) + x \end{array} \]

FPCore

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

C

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

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 + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 80.2%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 25.2

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

5. Applied rewrites 25.2%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 2.8%

   \[\leadsto x + \left(-x\right) \]

9. Final simplification 2.8%

   \[\leadsto \left(-x\right) + x \]
10. Add Preprocessing

Reproduce

herbie shell --seed 2024279 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))