Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 79.6% → 90.5%

Time: 10.8s

Alternatives: 17

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: 79.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \left(z - y\right) \cdot \frac{x - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-264}:\\ \;\;\;\;x - \frac{z - y}{\frac{a - z}{t - x}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{a}{z}, \frac{y - a}{z} \cdot \left(x - t\right), t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (- z y) (/ (- x t) (- z a))))))
   (if (<= t_1 -2e-264)
     (- x (/ (- z y) (/ (- a z) (- t x))))
     (if (<= t_1 0.0) (fma (+ 1.0 (/ a z)) (* (/ (- y a) z) (- x t)) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((z - y) * ((x - t) / (z - a)));
	double tmp;
	if (t_1 <= -2e-264) {
		tmp = x - ((z - y) / ((a - z) / (t - x)));
	} else if (t_1 <= 0.0) {
		tmp = fma((1.0 + (a / z)), (((y - a) / z) * (x - t)), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(z - y) * Float64(Float64(x - t) / Float64(z - a))))
	tmp = 0.0
	if (t_1 <= -2e-264)
		tmp = Float64(x - Float64(Float64(z - y) / Float64(Float64(a - z) / Float64(t - x))));
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(1.0 + Float64(a / z)), Float64(Float64(Float64(y - a) / z) * Float64(x - t)), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(z - y), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-264], N[(x - N[(N[(z - y), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(1.0 + N[(a / z), $MachinePrecision]), $MachinePrecision] * N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2e-264

   1. Initial program 92.3%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if -2e-264 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. +-commutative N/A

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

   5. Applied rewrites 99.0%

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 94.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(z - y\right) \cdot \frac{x - t}{z - a} \leq -2 \cdot 10^{-264}:\\ \;\;\;\;x - \frac{z - y}{\frac{a - z}{t - x}}\\ \mathbf{elif}\;x - \left(z - y\right) \cdot \frac{x - t}{z - a} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(1 + \frac{a}{z}, \frac{y - a}{z} \cdot \left(x - t\right), t\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(z - y\right) \cdot \frac{x - t}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \left(z - y\right) \cdot \frac{x - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-264}:\\ \;\;\;\;x - \frac{z - y}{\frac{a - z}{t - x}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (- z y) (/ (- x t) (- z a))))))
   (if (<= t_1 -2e-264)
     (- x (/ (- z y) (/ (- a z) (- t x))))
     (if (<= t_1 0.0) (fma (- x t) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((z - y) * ((x - t) / (z - a)));
	double tmp;
	if (t_1 <= -2e-264) {
		tmp = x - ((z - y) / ((a - z) / (t - x)));
	} else if (t_1 <= 0.0) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(z - y) * Float64(Float64(x - t) / Float64(z - a))))
	tmp = 0.0
	if (t_1 <= -2e-264)
		tmp = Float64(x - Float64(Float64(z - y) / Float64(Float64(a - z) / Float64(t - x))));
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(x - t), 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[(x - N[(N[(z - y), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-264], N[(x - N[(N[(z - y), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2e-264

   1. Initial program 92.3%

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

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

      4. un-div-inv N/A

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

      5. lower-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 92.3

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

   4. Applied rewrites 92.3%

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

   if -2e-264 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.6%

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

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 94.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(z - y\right) \cdot \frac{x - t}{z - a} \leq -2 \cdot 10^{-264}:\\ \;\;\;\;x - \frac{z - y}{\frac{a - z}{t - x}}\\ \mathbf{elif}\;x - \left(z - y\right) \cdot \frac{x - t}{z - a} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(z - y\right) \cdot \frac{x - t}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \left(z - y\right) \cdot \frac{x - t}{z - a}\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-264}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (- z y) (/ (- x t) (- z a))))))
   (if (<= t_1 -2e-264)
     t_1
     (if (<= t_1 0.0) (fma (- x t) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((z - y) * ((x - t) / (z - a)));
	double tmp;
	if (t_1 <= -2e-264) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(z - y) * Float64(Float64(x - t) / Float64(z - a))))
	tmp = 0.0
	if (t_1 <= -2e-264)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(x - t), 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[(x - N[(N[(z - y), $MachinePrecision] * N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-264], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(x - t), $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)))) < -2e-264 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 93.7%

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

   if -2e-264 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.6%

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

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(z - y\right) \cdot \frac{x - t}{z - a} \leq -2 \cdot 10^{-264}:\\ \;\;\;\;x - \left(z - y\right) \cdot \frac{x - t}{z - a}\\ \mathbf{elif}\;x - \left(z - y\right) \cdot \frac{x - t}{z - a} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(z - y\right) \cdot \frac{x - t}{z - a}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 90.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - t}{z - a}\\ t_2 := \mathsf{fma}\left(t\_1, y - z, x\right)\\ t_3 := x - \left(z - y\right) \cdot t\_1\\ \mathbf{if}\;t\_3 \leq -2 \cdot 10^{-264}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- x t) (- z a)))
        (t_2 (fma t_1 (- y z) x))
        (t_3 (- x (* (- z y) t_1))))
   (if (<= t_3 -2e-264)
     t_2
     (if (<= t_3 0.0) (fma (- x t) (/ (- y a) z) t) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (x - t) / (z - a);
	double t_2 = fma(t_1, (y - z), x);
	double t_3 = x - ((z - y) * t_1);
	double tmp;
	if (t_3 <= -2e-264) {
		tmp = t_2;
	} else if (t_3 <= 0.0) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(x - t) / Float64(z - a))
	t_2 = fma(t_1, Float64(y - z), x)
	t_3 = Float64(x - Float64(Float64(z - y) * t_1))
	tmp = 0.0
	if (t_3 <= -2e-264)
		tmp = t_2;
	elseif (t_3 <= 0.0)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t$95$1 * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$3 = N[(x - N[(N[(z - y), $MachinePrecision] * t$95$1), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -2e-264], t$95$2, If[LessEqual[t$95$3, 0.0], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2e-264 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 93.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 93.7

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 93.7

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

   4. Applied rewrites 93.7%

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

   if -2e-264 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.6%

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

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 98.9

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

   5. Applied rewrites 98.9%

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

4. Final simplification 94.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \left(z - y\right) \cdot \frac{x - t}{z - a} \leq -2 \cdot 10^{-264}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \mathbf{elif}\;x - \left(z - y\right) \cdot \frac{x - t}{z - a} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 71.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -6400000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;t - \frac{\left(-x\right) \cdot \left(y - a\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) a) (- t x) x)))
   (if (<= a -6400000.0)
     t_1
     (if (<= a 3.9e-159)
       (- t (/ (* (- t x) y) z))
       (if (<= a 2.15e-10) (- t (/ (* (- x) (- y a)) z)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / a), (t - x), x);
	double tmp;
	if (a <= -6400000.0) {
		tmp = t_1;
	} else if (a <= 3.9e-159) {
		tmp = t - (((t - x) * y) / z);
	} else if (a <= 2.15e-10) {
		tmp = t - ((-x * (y - a)) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -6400000.0)
		tmp = t_1;
	elseif (a <= 3.9e-159)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	elseif (a <= 2.15e-10)
		tmp = Float64(t - Float64(Float64(Float64(-x) * Float64(y - a)) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -6400000.0], t$95$1, If[LessEqual[a, 3.9e-159], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 2.15e-10], N[(t - N[(N[((-x) * N[(y - a), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.4e6 or 2.15000000000000007e-10 < a

   1. Initial program 94.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -6.4e6 < a < 3.89999999999999977e-159

   1. Initial program 77.4%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 80.4%

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

   if 3.89999999999999977e-159 < a < 2.15000000000000007e-10

   1. Initial program 64.6%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 59.9%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 72.2%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 72.3%

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

4. Final simplification 80.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6400000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 3.9 \cdot 10^{-159}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;t - \frac{\left(-x\right) \cdot \left(y - a\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 73.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -6400000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;t - \frac{\left(a - y\right) \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) a) (- t x) x)))
   (if (<= a -6400000.0)
     t_1
     (if (<= a 2.15e-10) (- t (/ (* (- a y) (- x t)) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / a), (t - x), x);
	double tmp;
	if (a <= -6400000.0) {
		tmp = t_1;
	} else if (a <= 2.15e-10) {
		tmp = t - (((a - y) * (x - t)) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -6400000.0)
		tmp = t_1;
	elseif (a <= 2.15e-10)
		tmp = Float64(t - Float64(Float64(Float64(a - y) * Float64(x - t)) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -6400000.0], t$95$1, If[LessEqual[a, 2.15e-10], N[(t - N[(N[(N[(a - y), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.4e6 or 2.15000000000000007e-10 < a

   1. Initial program 94.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -6.4e6 < a < 2.15000000000000007e-10

   1. Initial program 73.1%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 62.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 79.7%

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

4. Final simplification 81.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6400000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;t - \frac{\left(a - y\right) \cdot \left(x - t\right)}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -260000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) a) (- t x) x)))
   (if (<= a -260000000.0)
     t_1
     (if (<= a 2.15e-10) (fma (- x t) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / a), (t - x), x);
	double tmp;
	if (a <= -260000000.0) {
		tmp = t_1;
	} else if (a <= 2.15e-10) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -260000000.0)
		tmp = t_1;
	elseif (a <= 2.15e-10)
		tmp = fma(Float64(x - t), 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[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -260000000.0], t$95$1, If[LessEqual[a, 2.15e-10], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.6e8 or 2.15000000000000007e-10 < a

   1. Initial program 94.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   if -2.6e8 < a < 2.15000000000000007e-10

   1. Initial program 73.1%

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

   3. Taylor expanded in z around inf

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

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

      11. mul-1-neg N/A

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

      12. lower-neg.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower-/.f64 N/A

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

      15. lower--.f64 81.7

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

   5. Applied rewrites 81.7%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -260000000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -6400000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-74}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) a) (- t x) x)))
   (if (<= a -6400000.0)
     t_1
     (if (<= a 3.3e-74) (- t (/ (* (- t x) y) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / a), (t - x), x);
	double tmp;
	if (a <= -6400000.0) {
		tmp = t_1;
	} else if (a <= 3.3e-74) {
		tmp = t - (((t - x) * y) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -6400000.0)
		tmp = t_1;
	elseif (a <= 3.3e-74)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -6400000.0], t$95$1, If[LessEqual[a, 3.3e-74], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.4e6 or 3.29999999999999996e-74 < a

   1. Initial program 91.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 79.5

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

   5. Applied rewrites 79.5%

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

   if -6.4e6 < a < 3.29999999999999996e-74

   1. Initial program 73.3%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 77.6%

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

Alternative 9: 66.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{if}\;a \leq -215000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 4.2 \cdot 10^{-74}:\\ \;\;\;\;t - \frac{\left(t - x\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) a) y x)))
   (if (<= a -215000000.0)
     t_1
     (if (<= a 4.2e-74) (- t (/ (* (- t x) y) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / a), y, x);
	double tmp;
	if (a <= -215000000.0) {
		tmp = t_1;
	} else if (a <= 4.2e-74) {
		tmp = t - (((t - x) * y) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / a), y, x)
	tmp = 0.0
	if (a <= -215000000.0)
		tmp = t_1;
	elseif (a <= 4.2e-74)
		tmp = Float64(t - Float64(Float64(Float64(t - x) * y) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -215000000.0], t$95$1, If[LessEqual[a, 4.2e-74], N[(t - N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.15e8 or 4.2e-74 < a

   1. Initial program 91.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   if -2.15e8 < a < 4.2e-74

   1. Initial program 73.3%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 77.6%

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

Alternative 10: 61.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{if}\;a \leq -1.92 \cdot 10^{-81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-74}:\\ \;\;\;\;t - \frac{\left(-x\right) \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) a) y x)))
   (if (<= a -1.92e-81) t_1 (if (<= a 3.2e-74) (- t (/ (* (- x) y) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / a), y, x);
	double tmp;
	if (a <= -1.92e-81) {
		tmp = t_1;
	} else if (a <= 3.2e-74) {
		tmp = t - ((-x * y) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / a), y, x)
	tmp = 0.0
	if (a <= -1.92e-81)
		tmp = t_1;
	elseif (a <= 3.2e-74)
		tmp = Float64(t - Float64(Float64(Float64(-x) * y) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -1.92e-81], t$95$1, If[LessEqual[a, 3.2e-74], N[(t - N[(N[((-x) * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.92000000000000007e-81 or 3.1999999999999999e-74 < a

   1. Initial program 89.7%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 69.4

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

   5. Applied rewrites 69.4%

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

   if -1.92000000000000007e-81 < a < 3.1999999999999999e-74

   1. Initial program 73.8%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 66.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.3%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 70.9%

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

Alternative 11: 57.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{if}\;a \leq -6400000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-74}:\\ \;\;\;\;t - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) a) y x)))
   (if (<= a -6400000.0) t_1 (if (<= a 3.3e-74) (- t (/ (* t y) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / a), y, x);
	double tmp;
	if (a <= -6400000.0) {
		tmp = t_1;
	} else if (a <= 3.3e-74) {
		tmp = t - ((t * y) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / a), y, x)
	tmp = 0.0
	if (a <= -6400000.0)
		tmp = t_1;
	elseif (a <= 3.3e-74)
		tmp = Float64(t - Float64(Float64(t * y) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -6400000.0], t$95$1, If[LessEqual[a, 3.3e-74], N[(t - N[(N[(t * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.4e6 or 3.29999999999999996e-74 < a

   1. Initial program 91.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 72.3

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

   5. Applied rewrites 72.3%

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

   if -6.4e6 < a < 3.29999999999999996e-74

   1. Initial program 73.3%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 77.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 58.7%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6400000:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-74}:\\ \;\;\;\;t - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 58.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{if}\;a \leq -6400000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-74}:\\ \;\;\;\;t - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ y a) (- t x) x)))
   (if (<= a -6400000.0) t_1 (if (<= a 3.3e-74) (- t (/ (* t y) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y / a), (t - x), x);
	double tmp;
	if (a <= -6400000.0) {
		tmp = t_1;
	} else if (a <= 3.3e-74) {
		tmp = t - ((t * y) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y / a), Float64(t - x), x)
	tmp = 0.0
	if (a <= -6400000.0)
		tmp = t_1;
	elseif (a <= 3.3e-74)
		tmp = Float64(t - Float64(Float64(t * y) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -6400000.0], t$95$1, If[LessEqual[a, 3.3e-74], N[(t - N[(N[(t * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.4e6 or 3.29999999999999996e-74 < a

   1. Initial program 91.5%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 79.5

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

   5. Applied rewrites 79.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 72.3%

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

   if -6.4e6 < a < 3.29999999999999996e-74

   1. Initial program 73.3%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 64.3%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 77.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 58.7%

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

4. Final simplification 66.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -6400000:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{elif}\;a \leq 3.3 \cdot 10^{-74}:\\ \;\;\;\;t - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 53.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{if}\;a \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-36}:\\ \;\;\;\;t - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ t a) y x)))
   (if (<= a -2.8e+22) t_1 (if (<= a 6e-36) (- t (/ (* t y) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / a), y, x);
	double tmp;
	if (a <= -2.8e+22) {
		tmp = t_1;
	} else if (a <= 6e-36) {
		tmp = t - ((t * y) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / a), y, x)
	tmp = 0.0
	if (a <= -2.8e+22)
		tmp = t_1;
	elseif (a <= 6e-36)
		tmp = Float64(t - Float64(Float64(t * y) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -2.8e+22], t$95$1, If[LessEqual[a, 6e-36], N[(t - N[(N[(t * y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.8e22 or 6.0000000000000003e-36 < a

   1. Initial program 91.7%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 72.6

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

   5. Applied rewrites 72.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 65.2%

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

   if -2.8e22 < a < 6.0000000000000003e-36

   1. Initial program 74.5%

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

   3. Taylor expanded in z around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 61.1%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 75.1%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 56.2%

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

4. Final simplification 61.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.8 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;a \leq 6 \cdot 10^{-36}:\\ \;\;\;\;t - \frac{t \cdot y}{z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 47.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) + x\\ \mathbf{if}\;z \leq -7.6 \cdot 10^{+173}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- t x) x)))
   (if (<= z -7.6e+173) t_1 (if (<= z 1.4e+142) (fma (/ t a) y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) + x;
	double tmp;
	if (z <= -7.6e+173) {
		tmp = t_1;
	} else if (z <= 1.4e+142) {
		tmp = fma((t / a), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) + x)
	tmp = 0.0
	if (z <= -7.6e+173)
		tmp = t_1;
	elseif (z <= 1.4e+142)
		tmp = fma(Float64(t / a), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -7.6e+173], t$95$1, If[LessEqual[z, 1.4e+142], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.60000000000000022e173 or 1.4e142 < z

   1. Initial program 63.8%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 51.7

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

   5. Applied rewrites 51.7%

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

   if -7.60000000000000022e173 < z < 1.4e142

   1. Initial program 89.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 65.9

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

   5. Applied rewrites 65.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 55.2%

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

4. Final simplification 54.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{+173}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{elif}\;z \leq 1.4 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 28.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) + x\\ \mathbf{if}\;z \leq -6 \cdot 10^{+166}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+56}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- t x) x)))
   (if (<= z -6e+166) t_1 (if (<= z 1.95e+56) (* (/ y a) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - x) + x;
	double tmp;
	if (z <= -6e+166) {
		tmp = t_1;
	} else if (z <= 1.95e+56) {
		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 = (t - x) + x
    if (z <= (-6d+166)) then
        tmp = t_1
    else if (z <= 1.95d+56) 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 = (t - x) + x;
	double tmp;
	if (z <= -6e+166) {
		tmp = t_1;
	} else if (z <= 1.95e+56) {
		tmp = (y / a) * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (t - x) + x
	tmp = 0
	if z <= -6e+166:
		tmp = t_1
	elif z <= 1.95e+56:
		tmp = (y / a) * t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - x) + x)
	tmp = 0.0
	if (z <= -6e+166)
		tmp = t_1;
	elseif (z <= 1.95e+56)
		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 = (t - x) + x;
	tmp = 0.0;
	if (z <= -6e+166)
		tmp = t_1;
	elseif (z <= 1.95e+56)
		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[(N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -6e+166], t$95$1, If[LessEqual[z, 1.95e+56], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.99999999999999997e166 or 1.94999999999999997e56 < z

   1. Initial program 69.4%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 42.0

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

   5. Applied rewrites 42.0%

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

   if -5.99999999999999997e166 < z < 1.94999999999999997e56

   1. Initial program 90.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 69.1

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

   5. Applied rewrites 69.1%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 21.3%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 28.5%

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

4. Final simplification 32.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{+166}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+56}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 18.7% 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 83.8%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
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 18.6

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

5. Applied rewrites 18.6%

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

6. Final simplification 18.6%

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

Alternative 17: 2.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 83.8%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
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 18.6

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

5. Applied rewrites 18.6%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 2.6%

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

9. Final simplification 2.6%

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

Reproduce

herbie shell --seed 2024235 
(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)))))