Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 79.8% → 94.1%

Time: 11.2s

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: 79.8% 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: 94.1% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)\\ t_2 := x - \frac{x - t}{a - z} \cdot \left(y - z\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) (- z a)) (- t x) x))
        (t_2 (- x (* (/ (- x t) (- a z)) (- y z)))))
   (if (<= t_2 -5e-290)
     t_1
     (if (<= t_2 5e-258) (fma (fma t -1.0 x) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / (z - a)), (t - x), x);
	double t_2 = x - (((x - t) / (a - z)) * (y - z));
	double tmp;
	if (t_2 <= -5e-290) {
		tmp = t_1;
	} else if (t_2 <= 5e-258) {
		tmp = fma(fma(t, -1.0, x), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / Float64(z - a)), Float64(t - x), x)
	t_2 = Float64(x - Float64(Float64(Float64(x - t) / Float64(a - z)) * Float64(y - z)))
	tmp = 0.0
	if (t_2 <= -5e-290)
		tmp = t_1;
	elseif (t_2 <= 5e-258)
		tmp = fma(fma(t, -1.0, x), Float64(Float64(y - a) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(N[(x - t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-290], t$95$1, If[LessEqual[t$95$2, 5e-258], N[(N[(t * -1.0 + x), $MachinePrecision] * N[(N[(y - a), $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-290 or 4.9999999999999999e-258 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 90.8%

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

      1. lift-+.f64 N/A

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

      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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 94.9

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

   4. Applied rewrites 94.9%

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

   if -5.0000000000000001e-290 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.9999999999999999e-258

   1. Initial program 4.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 96.4%

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

4. Final simplification 95.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{x - t}{a - z} \cdot \left(y - z\right) \leq -5 \cdot 10^{-290}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)\\ \mathbf{elif}\;x - \frac{x - t}{a - z} \cdot \left(y - z\right) \leq 5 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z - a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ t_2 := x - \frac{x - t}{a - z} \cdot \left(y - z\right)\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-290}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) (- z a)) (- y z) x))
        (t_2 (- x (* (/ (- x t) (- a z)) (- y z)))))
   (if (<= t_2 -5e-290)
     t_1
     (if (<= t_2 5e-258) (fma (fma t -1.0 x) (/ (- y a) 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 = x - (((x - t) / (a - z)) * (y - z));
	double tmp;
	if (t_2 <= -5e-290) {
		tmp = t_1;
	} else if (t_2 <= 5e-258) {
		tmp = fma(fma(t, -1.0, x), ((y - a) / 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(x - Float64(Float64(Float64(x - t) / Float64(a - z)) * Float64(y - z)))
	tmp = 0.0
	if (t_2 <= -5e-290)
		tmp = t_1;
	elseif (t_2 <= 5e-258)
		tmp = fma(fma(t, -1.0, x), Float64(Float64(y - a) / 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[(x - N[(N[(N[(x - t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-290], t$95$1, If[LessEqual[t$95$2, 5e-258], N[(N[(t * -1.0 + x), $MachinePrecision] * N[(N[(y - a), $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-290 or 4.9999999999999999e-258 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 90.8%

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

      1. lift-+.f64 N/A

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

      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 90.8

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

      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 90.8

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

   4. Applied rewrites 90.8%

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

   if -5.0000000000000001e-290 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.9999999999999999e-258

   1. Initial program 4.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 96.4%

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

4. Final simplification 91.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{x - t}{a - z} \cdot \left(y - z\right) \leq -5 \cdot 10^{-290}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \mathbf{elif}\;x - \frac{x - t}{a - z} \cdot \left(y - z\right) \leq 5 \cdot 10^{-258}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+226}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+76}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4.6e+226)
   (* (/ (- z y) z) t)
   (if (<= z -4e+76)
     (* (- x t) (/ y (- z a)))
     (if (<= z 7.5e+69) (fma (/ y a) (- t x) x) (* (/ t (- z a)) (- z y))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.6e+226) {
		tmp = ((z - y) / z) * t;
	} else if (z <= -4e+76) {
		tmp = (x - t) * (y / (z - a));
	} else if (z <= 7.5e+69) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = (t / (z - a)) * (z - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.6e+226)
		tmp = Float64(Float64(Float64(z - y) / z) * t);
	elseif (z <= -4e+76)
		tmp = Float64(Float64(x - t) * Float64(y / Float64(z - a)));
	elseif (z <= 7.5e+69)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = Float64(Float64(t / Float64(z - a)) * Float64(z - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.6e+226], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, -4e+76], N[(N[(x - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.5e+69], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.5999999999999999e226

   1. Initial program 41.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 30.4

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

   5. Applied rewrites 30.4%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 57.2%

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

   if -4.5999999999999999e226 < z < -4.0000000000000002e76

   1. Initial program 72.1%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 47.7

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

   5. Applied rewrites 47.7%

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

   if -4.0000000000000002e76 < z < 7.49999999999999939e69

   1. Initial program 90.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 94.2

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 79.0

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

   7. Applied rewrites 79.0%

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

   if 7.49999999999999939e69 < z

   1. Initial program 75.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 67.8

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

   5. Applied rewrites 67.8%

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

4. Final simplification 71.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+226}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+76}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 61.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - y}{z} \cdot t\\ \mathbf{if}\;z \leq -4.6 \cdot 10^{+226}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+76}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z y) z) t)))
   (if (<= z -4.6e+226)
     t_1
     (if (<= z -4e+76)
       (* (- x t) (/ y (- z a)))
       (if (<= z 8.2e+69) (fma (/ y a) (- t x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - y) / z) * t;
	double tmp;
	if (z <= -4.6e+226) {
		tmp = t_1;
	} else if (z <= -4e+76) {
		tmp = (x - t) * (y / (z - a));
	} else if (z <= 8.2e+69) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - y) / z) * t)
	tmp = 0.0
	if (z <= -4.6e+226)
		tmp = t_1;
	elseif (z <= -4e+76)
		tmp = Float64(Float64(x - t) * Float64(y / Float64(z - a)));
	elseif (z <= 8.2e+69)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -4.6e+226], t$95$1, If[LessEqual[z, -4e+76], N[(N[(x - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 8.2e+69], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.5999999999999999e226 or 8.1999999999999998e69 < z

   1. Initial program 67.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 58.0

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

   5. Applied rewrites 58.0%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 63.7%

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

   if -4.5999999999999999e226 < z < -4.0000000000000002e76

   1. Initial program 72.1%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 47.7

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

   5. Applied rewrites 47.7%

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

   if -4.0000000000000002e76 < z < 8.1999999999999998e69

   1. Initial program 90.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 94.2

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 79.0

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

   7. Applied rewrites 79.0%

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

4. Final simplification 70.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.6 \cdot 10^{+226}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+76}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z - a}\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 78.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{if}\;z \leq -7.2 \cdot 10^{+145}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+109}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (fma t -1.0 x) (/ (- y a) z) t)))
   (if (<= z -7.2e+145)
     t_1
     (if (<= z 3.05e+109) (+ (* (- x t) (/ y (- z a))) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(fma(t, -1.0, x), ((y - a) / z), t);
	double tmp;
	if (z <= -7.2e+145) {
		tmp = t_1;
	} else if (z <= 3.05e+109) {
		tmp = ((x - t) * (y / (z - a))) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(fma(t, -1.0, x), Float64(Float64(y - a) / z), t)
	tmp = 0.0
	if (z <= -7.2e+145)
		tmp = t_1;
	elseif (z <= 3.05e+109)
		tmp = Float64(Float64(Float64(x - t) * Float64(y / Float64(z - a))) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t * -1.0 + x), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -7.2e+145], t$95$1, If[LessEqual[z, 3.05e+109], N[(N[(N[(x - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.19999999999999948e145 or 3.05000000000000004e109 < z

   1. Initial program 61.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 81.1%

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

   if -7.19999999999999948e145 < z < 3.05000000000000004e109

   1. Initial program 89.2%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

4. Final simplification 82.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+145}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{elif}\;z \leq 3.05 \cdot 10^{+109}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 72.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+145}:\\ \;\;\;\;t - \frac{\left(a - y\right) \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+112}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -7.2e+145)
   (- t (/ (* (- a y) (- x t)) z))
   (if (<= z 1.9e+112) (+ (* (- x t) (/ y (- z a))) x) (* (/ (- z y) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -7.2e+145) {
		tmp = t - (((a - y) * (x - t)) / z);
	} else if (z <= 1.9e+112) {
		tmp = ((x - t) * (y / (z - a))) + x;
	} else {
		tmp = ((z - y) / z) * 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 (z <= (-7.2d+145)) then
        tmp = t - (((a - y) * (x - t)) / z)
    else if (z <= 1.9d+112) then
        tmp = ((x - t) * (y / (z - a))) + x
    else
        tmp = ((z - y) / z) * 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 (z <= -7.2e+145) {
		tmp = t - (((a - y) * (x - t)) / z);
	} else if (z <= 1.9e+112) {
		tmp = ((x - t) * (y / (z - a))) + x;
	} else {
		tmp = ((z - y) / z) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -7.2e+145:
		tmp = t - (((a - y) * (x - t)) / z)
	elif z <= 1.9e+112:
		tmp = ((x - t) * (y / (z - a))) + x
	else:
		tmp = ((z - y) / z) * t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -7.2e+145)
		tmp = Float64(t - Float64(Float64(Float64(a - y) * Float64(x - t)) / z));
	elseif (z <= 1.9e+112)
		tmp = Float64(Float64(Float64(x - t) * Float64(y / Float64(z - a))) + x);
	else
		tmp = Float64(Float64(Float64(z - y) / z) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -7.2e+145)
		tmp = t - (((a - y) * (x - t)) / z);
	elseif (z <= 1.9e+112)
		tmp = ((x - t) * (y / (z - a))) + x;
	else
		tmp = ((z - y) / z) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -7.2e+145], N[(t - N[(N[(N[(a - y), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.9e+112], N[(N[(N[(x - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -7.19999999999999948e145

   1. Initial program 52.2%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 60.9

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

   4. Applied rewrites 60.9%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 67.7%

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

   if -7.19999999999999948e145 < z < 1.90000000000000004e112

   1. Initial program 89.2%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 83.5

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

   5. Applied rewrites 83.5%

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

   if 1.90000000000000004e112 < z

   1. Initial program 69.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 73.4%

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

4. Final simplification 79.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.2 \cdot 10^{+145}:\\ \;\;\;\;t - \frac{\left(a - y\right) \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;z \leq 1.9 \cdot 10^{+112}:\\ \;\;\;\;\left(x - t\right) \cdot \frac{y}{z - a} + x\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 65.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+120}:\\ \;\;\;\;t - \frac{\left(a - y\right) \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -8.8e+120)
   (- t (/ (* (- a y) (- x t)) z))
   (if (<= z 7.5e+69)
     (fma (/ (- y z) a) (- t x) x)
     (* (/ t (- z a)) (- z y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -8.8e+120) {
		tmp = t - (((a - y) * (x - t)) / z);
	} else if (z <= 7.5e+69) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else {
		tmp = (t / (z - a)) * (z - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -8.8e+120)
		tmp = Float64(t - Float64(Float64(Float64(a - y) * Float64(x - t)) / z));
	elseif (z <= 7.5e+69)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	else
		tmp = Float64(Float64(t / Float64(z - a)) * Float64(z - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -8.8e+120], N[(t - N[(N[(N[(a - y), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.5e+69], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -8.8000000000000005e120

   1. Initial program 53.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 60.8

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

   4. Applied rewrites 60.8%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r/ N/A

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

      4. mul-1-neg N/A

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

      5. div-sub N/A

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

      6. mul-1-neg N/A

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

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

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

      8. associate-*r/ N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 N/A

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

   7. Applied rewrites 66.8%

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

   if -8.8000000000000005e120 < z < 7.49999999999999939e69

   1. Initial program 89.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 78.4

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

   5. Applied rewrites 78.4%

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

   if 7.49999999999999939e69 < z

   1. Initial program 75.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 67.8

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

   5. Applied rewrites 67.8%

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

4. Final simplification 74.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.8 \cdot 10^{+120}:\\ \;\;\;\;t - \frac{\left(a - y\right) \cdot \left(x - t\right)}{z}\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 62.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.6e+246)
   (* (/ (- z y) z) t)
   (if (<= z 7.5e+69)
     (fma (/ (- y z) a) (- t x) x)
     (* (/ t (- z a)) (- z y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.6e+246) {
		tmp = ((z - y) / z) * t;
	} else if (z <= 7.5e+69) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else {
		tmp = (t / (z - a)) * (z - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.6e+246)
		tmp = Float64(Float64(Float64(z - y) / z) * t);
	elseif (z <= 7.5e+69)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	else
		tmp = Float64(Float64(t / Float64(z - a)) * Float64(z - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.6e+246], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 7.5e+69], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.6e246

   1. Initial program 20.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 28.2

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

   5. Applied rewrites 28.2%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.4%

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

   if -9.6e246 < z < 7.49999999999999939e69

   1. Initial program 86.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 73.2

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

   5. Applied rewrites 73.2%

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

   if 7.49999999999999939e69 < z

   1. Initial program 75.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 67.8

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

   5. Applied rewrites 67.8%

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

4. Final simplification 71.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 63.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - y}{z} \cdot t\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- z y) z) t)))
   (if (<= z -2.2e+92) t_1 (if (<= z 8.2e+69) (fma (/ y a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((z - y) / z) * t;
	double tmp;
	if (z <= -2.2e+92) {
		tmp = t_1;
	} else if (z <= 8.2e+69) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(z - y) / z) * t)
	tmp = 0.0
	if (z <= -2.2e+92)
		tmp = t_1;
	elseif (z <= 8.2e+69)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -2.2e+92], t$95$1, If[LessEqual[z, 8.2e+69], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.19999999999999992e92 or 8.1999999999999998e69 < z

   1. Initial program 68.1%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 53.7

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

   5. Applied rewrites 53.7%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 55.2%

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

   if -2.19999999999999992e92 < z < 8.1999999999999998e69

   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 94.1

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

   4. Applied rewrites 94.1%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 77.2

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

   7. Applied rewrites 77.2%

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

4. Final simplification 68.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{+92}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \mathbf{elif}\;z \leq 8.2 \cdot 10^{+69}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z - y}{z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 58.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{z - a} \cdot t\\ \mathbf{if}\;z \leq -9.6 \cdot 10^{+246}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ z (- z a)) t)))
   (if (<= z -9.6e+246) t_1 (if (<= z 4.1e+109) (fma (/ y a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z / (z - a)) * t;
	double tmp;
	if (z <= -9.6e+246) {
		tmp = t_1;
	} else if (z <= 4.1e+109) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z / Float64(z - a)) * t)
	tmp = 0.0
	if (z <= -9.6e+246)
		tmp = t_1;
	elseif (z <= 4.1e+109)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[z, -9.6e+246], t$95$1, If[LessEqual[z, 4.1e+109], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.6e246 or 4.0999999999999997e109 < z

   1. Initial program 58.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 61.2

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

   5. Applied rewrites 61.2%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 65.2%

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

   if -9.6e246 < z < 4.0999999999999997e109

   1. Initial program 87.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 67.6

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

   7. Applied rewrites 67.6%

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

4. Final simplification 67.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{z}{z - a} \cdot t\\ \mathbf{elif}\;z \leq 4.1 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - a} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 55.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{-t}{x} \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{-z} \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.6e+246)
   (* (/ (- t) x) (- x))
   (if (<= z 1.12e+110) (fma (/ y a) (- t x) x) (* (/ t (- z)) (- z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.6e+246) {
		tmp = (-t / x) * -x;
	} else if (z <= 1.12e+110) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = (t / -z) * -z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.6e+246)
		tmp = Float64(Float64(Float64(-t) / x) * Float64(-x));
	elseif (z <= 1.12e+110)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = Float64(Float64(t / Float64(-z)) * Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.6e+246], N[(N[((-t) / x), $MachinePrecision] * (-x)), $MachinePrecision], If[LessEqual[z, 1.12e+110], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / (-z)), $MachinePrecision] * (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.6e246

   1. Initial program 20.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. sub-neg N/A

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 60.7%

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

   if -9.6e246 < z < 1.1200000000000001e110

   1. Initial program 87.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 67.6

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

   7. Applied rewrites 67.6%

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

   if 1.1200000000000001e110 < z

   1. Initial program 69.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 59.2%

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

4. Final simplification 66.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{-t}{x} \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{-z} \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{-t}{x} \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{-z} \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.6e+246)
   (* (/ (- t) x) (- x))
   (if (<= z 1.12e+110) (fma (/ (- t x) a) y x) (* (/ t (- z)) (- z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.6e+246) {
		tmp = (-t / x) * -x;
	} else if (z <= 1.12e+110) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = (t / -z) * -z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.6e+246)
		tmp = Float64(Float64(Float64(-t) / x) * Float64(-x));
	elseif (z <= 1.12e+110)
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = Float64(Float64(t / Float64(-z)) * Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.6e+246], N[(N[((-t) / x), $MachinePrecision] * (-x)), $MachinePrecision], If[LessEqual[z, 1.12e+110], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(t / (-z)), $MachinePrecision] * (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.6e246

   1. Initial program 20.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. sub-neg N/A

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 60.7%

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

   if -9.6e246 < z < 1.1200000000000001e110

   1. Initial program 87.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 66.7

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

   5. Applied rewrites 66.7%

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

   if 1.1200000000000001e110 < z

   1. Initial program 69.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 59.2%

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

4. Final simplification 65.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{-t}{x} \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+110}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{-z} \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 46.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{-t}{x} \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{-z} \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.6e+246)
   (* (/ (- t) x) (- x))
   (if (<= z 4.2e+109) (fma y (/ t a) x) (* (/ t (- z)) (- z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.6e+246) {
		tmp = (-t / x) * -x;
	} else if (z <= 4.2e+109) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = (t / -z) * -z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.6e+246)
		tmp = Float64(Float64(Float64(-t) / x) * Float64(-x));
	elseif (z <= 4.2e+109)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(Float64(t / Float64(-z)) * Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.6e+246], N[(N[((-t) / x), $MachinePrecision] * (-x)), $MachinePrecision], If[LessEqual[z, 4.2e+109], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / (-z)), $MachinePrecision] * (-z)), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.6e246

   1. Initial program 20.4%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. sub-neg N/A

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 60.7%

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

   if -9.6e246 < z < 4.2000000000000003e109

   1. Initial program 87.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 91.4

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

   4. Applied rewrites 91.4%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 66.7

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

   7. Applied rewrites 66.7%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 55.5%

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

   if 4.2000000000000003e109 < z

   1. Initial program 69.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 59.2%

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

4. Final simplification 56.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.6 \cdot 10^{+246}:\\ \;\;\;\;\frac{-t}{x} \cdot \left(-x\right)\\ \mathbf{elif}\;z \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{-z} \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 41.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+210}:\\ \;\;\;\;\frac{y}{z - a} \cdot x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= y -1.4e+210)
   (* (/ y (- z a)) x)
   (if (<= y 2.5e+92) (fma y (/ t a) x) (* (/ y (- a z)) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.4e+210) {
		tmp = (y / (z - a)) * x;
	} else if (y <= 2.5e+92) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = (y / (a - z)) * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.4e+210)
		tmp = Float64(Float64(y / Float64(z - a)) * x);
	elseif (y <= 2.5e+92)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(Float64(y / Float64(a - z)) * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.4e+210], N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[y, 2.5e+92], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.4000000000000001e210

   1. Initial program 84.3%

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

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

      4. frac-2neg N/A

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

      5. lower-/.f64 N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-*.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 55.7

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

   4. Applied rewrites 55.7%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 N/A

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

      6. div-sub N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 61.3

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

   7. Applied rewrites 61.3%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 72.3%

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

   if -1.4000000000000001e210 < y < 2.50000000000000011e92

   1. Initial program 80.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 84.3

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

   4. Applied rewrites 84.3%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 57.1

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

   7. Applied rewrites 57.1%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 51.5%

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

   if 2.50000000000000011e92 < y

   1. Initial program 86.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 58.3

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

   5. Applied rewrites 58.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 55.6%

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

4. Final simplification 53.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.4 \cdot 10^{+210}:\\ \;\;\;\;\frac{y}{z - a} \cdot x\\ \mathbf{elif}\;y \leq 2.5 \cdot 10^{+92}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 29.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \left(-x\right)\\ \mathbf{if}\;a \leq -1.82 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* -1.0 (- x))))
   (if (<= a -1.82e+155) t_1 (if (<= a 1.8e+67) (* (/ y a) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -1.0 * -x;
	double tmp;
	if (a <= -1.82e+155) {
		tmp = t_1;
	} else if (a <= 1.8e+67) {
		tmp = (y / a) * 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 = (-1.0d0) * -x
    if (a <= (-1.82d+155)) then
        tmp = t_1
    else if (a <= 1.8d+67) then
        tmp = (y / a) * 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 = -1.0 * -x;
	double tmp;
	if (a <= -1.82e+155) {
		tmp = t_1;
	} else if (a <= 1.8e+67) {
		tmp = (y / a) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -1.0 * -x
	tmp = 0
	if a <= -1.82e+155:
		tmp = t_1
	elif a <= 1.8e+67:
		tmp = (y / a) * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(-1.0 * Float64(-x))
	tmp = 0.0
	if (a <= -1.82e+155)
		tmp = t_1;
	elseif (a <= 1.8e+67)
		tmp = Float64(Float64(y / a) * t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = -1.0 * -x;
	tmp = 0.0;
	if (a <= -1.82e+155)
		tmp = t_1;
	elseif (a <= 1.8e+67)
		tmp = (y / a) * t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(-1.0 * (-x)), $MachinePrecision]}, If[LessEqual[a, -1.82e+155], t$95$1, If[LessEqual[a, 1.8e+67], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.81999999999999989e155 or 1.7999999999999999e67 < a

   1. Initial program 90.0%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. sub-neg N/A

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

   5. Applied rewrites 83.5%

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

   6. Taylor expanded in a around inf

      \[\leadsto \left(-x\right) \cdot -1 \]
   7. Step-by-step derivation

   8. Applied rewrites 61.3%

      \[\leadsto \left(-x\right) \cdot -1 \]

   if -1.81999999999999989e155 < a < 1.7999999999999999e67

   1. Initial program 76.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 54.1

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

   5. Applied rewrites 54.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 29.4%

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

4. Final simplification 40.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.82 \cdot 10^{+155}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \mathbf{elif}\;a \leq 1.8 \cdot 10^{+67}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 44.1% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{-z} \cdot \left(-z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 4.2e+109) (fma y (/ t a) x) (* (/ t (- z)) (- z))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 4.2e+109) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = (t / -z) * -z;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 4.2e+109)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(Float64(t / Float64(-z)) * Float64(-z));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 4.2e+109], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(t / (-z)), $MachinePrecision] * (-z)), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 4.2000000000000003e109

   1. Initial program 83.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 63.2

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

   7. Applied rewrites 63.2%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 52.7%

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

   if 4.2000000000000003e109 < z

   1. Initial program 69.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 71.1

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

   5. Applied rewrites 71.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 68.6%

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

   9. Taylor expanded in z around inf

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

   11. Applied rewrites 59.2%

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

4. Final simplification 53.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 4.2 \cdot 10^{+109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{-z} \cdot \left(-z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 43.6% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq 5.6 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z 5.6e+112) (fma y (/ t a) x) (+ (- t x) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 5.6e+112) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = (t - x) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 5.6e+112)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = Float64(Float64(t - x) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 5.6e+112], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 5.6000000000000003e112

   1. Initial program 83.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-*r/ N/A

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

      7. div-inv N/A

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

      8. times-frac N/A

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

      9. lift--.f64 N/A

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

      10. flip-- N/A

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

      11. clear-num N/A

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

      12. clear-num N/A

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

      13. flip-- N/A

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

      14. lift--.f64 N/A

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

      15. lower-fma.f64 N/A

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

      16. lower-/.f64 88.7

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

   4. Applied rewrites 88.7%

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

   5. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 63.2

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

   7. Applied rewrites 63.2%

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

   8. Taylor expanded in t around inf

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

   10. Applied rewrites 52.7%

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

   if 5.6000000000000003e112 < z

   1. Initial program 69.3%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 45.4

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

   5. Applied rewrites 45.4%

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

4. Final simplification 51.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq 5.6 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 18: 32.9% accurate, 1.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := -1 \cdot \left(-x\right)\\ \mathbf{if}\;a \leq -3.4 \cdot 10^{+100}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 64000000000000:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* -1.0 (- x))))
   (if (<= a -3.4e+100) t_1 (if (<= a 64000000000000.0) (+ (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = -1.0 * -x;
	double tmp;
	if (a <= -3.4e+100) {
		tmp = t_1;
	} else if (a <= 64000000000000.0) {
		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 = (-1.0d0) * -x
    if (a <= (-3.4d+100)) then
        tmp = t_1
    else if (a <= 64000000000000.0d0) 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 = -1.0 * -x;
	double tmp;
	if (a <= -3.4e+100) {
		tmp = t_1;
	} else if (a <= 64000000000000.0) {
		tmp = (t - x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = -1.0 * -x
	tmp = 0
	if a <= -3.4e+100:
		tmp = t_1
	elif a <= 64000000000000.0:
		tmp = (t - x) + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(-1.0 * Float64(-x))
	tmp = 0.0
	if (a <= -3.4e+100)
		tmp = t_1;
	elseif (a <= 64000000000000.0)
		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 = -1.0 * -x;
	tmp = 0.0;
	if (a <= -3.4e+100)
		tmp = t_1;
	elseif (a <= 64000000000000.0)
		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[(-1.0 * (-x)), $MachinePrecision]}, If[LessEqual[a, -3.4e+100], t$95$1, If[LessEqual[a, 64000000000000.0], N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -3.39999999999999994e100 or 6.4e13 < a

   1. Initial program 88.6%

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

   3. Taylor expanded in x around -inf

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

      1. mul-1-neg N/A

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

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

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

      3. lower-*.f64 N/A

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

      4. lower-neg.f64 N/A

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

      5. +-commutative N/A

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

      6. associate--r+ N/A

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

      7. sub-neg N/A

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

   5. Applied rewrites 82.2%

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

   6. Taylor expanded in a around inf

      \[\leadsto \left(-x\right) \cdot -1 \]
   7. Step-by-step derivation

   8. Applied rewrites 55.1%

      \[\leadsto \left(-x\right) \cdot -1 \]

   if -3.39999999999999994e100 < a < 6.4e13

   1. Initial program 76.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 21.6

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

   5. Applied rewrites 21.6%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3.4 \cdot 10^{+100}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \mathbf{elif}\;a \leq 64000000000000:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;-1 \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 19: 18.9% 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 81.3%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
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 16.6

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

5. Applied rewrites 16.6%

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

6. Final simplification 16.6%

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

Alternative 20: 2.8% accurate, 29.0× speedup

Math

\[\begin{array}{l} \\ 0 \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 0.0)

C

double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

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 = 0.0d0
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return 0.0;
}

Python

def code(x, y, z, t, a):
	return 0.0

Julia

function code(x, y, z, t, a)
	return 0.0
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = 0.0;
end

Wolfram

code[x_, y_, z_, t_, a_] := 0.0

Derivation

1. Initial program 81.3%

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

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

   4. frac-2neg N/A

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

   5. lower-/.f64 N/A

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

   6. *-commutative N/A

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

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

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

   8. lower-*.f64 N/A

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

   9. neg-sub0 N/A

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

   10. lift--.f64 N/A

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

   11. sub-neg N/A

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

   12. +-commutative N/A

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

   13. associate--r+ N/A

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

   14. neg-sub0 N/A

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

   15. remove-double-neg N/A

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

   16. lower--.f64 N/A

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

   17. neg-sub0 N/A

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

   18. lift--.f64 N/A

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

   19. sub-neg N/A

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

   20. +-commutative N/A

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

   21. associate--r+ N/A

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

   22. neg-sub0 N/A

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

   23. remove-double-neg N/A

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

   24. lower--.f64 69.2

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

4. Applied rewrites 69.2%

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

5. Taylor expanded in t around 0

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

   1. +-commutative N/A

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

   2. associate-/l* N/A

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

   3. div-sub N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 N/A

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

   6. div-sub N/A

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

   7. lower-/.f64 N/A

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

   8. lower--.f64 N/A

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

   9. lower--.f64 45.7

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

7. Applied rewrites 45.7%

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

8. Taylor expanded in z around inf

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

10. Applied rewrites 2.8%

    \[\leadsto 0 \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2024255 
(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)))))