Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 80.6% → 91.3%

Time: 11.3s

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: 80.6% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - z}{\frac{z - a}{x - t}} + x\\ t_2 := \frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (/ (- y z) (/ (- z a) (- x t))) x))
        (t_2 (+ (* (/ (- t x) (- a z)) (- y z)) x)))
   (if (<= t_2 -2e-255)
     t_1
     (if (<= t_2 5e-278) (fma (fma t -1.0 x) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - z) / ((z - a) / (x - t))) + x;
	double t_2 = (((t - x) / (a - z)) * (y - z)) + x;
	double tmp;
	if (t_2 <= -2e-255) {
		tmp = t_1;
	} else if (t_2 <= 5e-278) {
		tmp = fma(fma(t, -1.0, x), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y - z) / Float64(Float64(z - a) / Float64(x - t))) + x)
	t_2 = Float64(Float64(Float64(Float64(t - x) / Float64(a - z)) * Float64(y - z)) + x)
	tmp = 0.0
	if (t_2 <= -2e-255)
		tmp = t_1;
	elseif (t_2 <= 5e-278)
		tmp = fma(fma(t, -1.0, x), Float64(Float64(y - a) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / N[(N[(z - a), $MachinePrecision] / N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-255], t$95$1, If[LessEqual[t$95$2, 5e-278], N[(N[(t * -1.0 + x), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2e-255 or 4.99999999999999985e-278 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 91.1%

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

      1. lift-*.f64 N/A

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

      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 91.2

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

   4. Applied rewrites 91.2%

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

   if -2e-255 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.99999999999999985e-278

   1. Initial program 6.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 93.4%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq -2 \cdot 10^{-255}:\\ \;\;\;\;\frac{y - z}{\frac{z - a}{x - t}} + x\\ \mathbf{elif}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq 5 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{y - z}{\frac{z - a}{x - t}} + x\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* (/ (- t x) (- a z)) (- y z)) x)))
   (if (<= t_1 -2e-255)
     t_1
     (if (<= t_1 5e-278) (fma (fma t -1.0 x) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (((t - x) / (a - z)) * (y - z)) + x;
	double tmp;
	if (t_1 <= -2e-255) {
		tmp = t_1;
	} else if (t_1 <= 5e-278) {
		tmp = fma(fma(t, -1.0, x), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(Float64(t - x) / Float64(a - z)) * Float64(y - z)) + x)
	tmp = 0.0
	if (t_1 <= -2e-255)
		tmp = t_1;
	elseif (t_1 <= 5e-278)
		tmp = fma(fma(t, -1.0, x), Float64(Float64(y - a) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-255], t$95$1, If[LessEqual[t$95$1, 5e-278], N[(N[(t * -1.0 + x), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2e-255 or 4.99999999999999985e-278 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 91.1%

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

   if -2e-255 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.99999999999999985e-278

   1. Initial program 6.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 93.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq -2 \cdot 10^{-255}:\\ \;\;\;\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \mathbf{elif}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq 5 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 91.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ t_2 := \frac{t - x}{a - z} \cdot \left(y - z\right) + x\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-255}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) (- z a)) (- y z) x))
        (t_2 (+ (* (/ (- t x) (- a z)) (- y z)) x)))
   (if (<= t_2 -2e-255)
     t_1
     (if (<= t_2 5e-278) (fma (fma t -1.0 x) (/ (- y a) z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / (z - a)), (y - z), x);
	double t_2 = (((t - x) / (a - z)) * (y - z)) + x;
	double tmp;
	if (t_2 <= -2e-255) {
		tmp = t_1;
	} else if (t_2 <= 5e-278) {
		tmp = fma(fma(t, -1.0, x), ((y - a) / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / Float64(z - a)), Float64(y - z), x)
	t_2 = Float64(Float64(Float64(Float64(t - x) / Float64(a - z)) * Float64(y - z)) + x)
	tmp = 0.0
	if (t_2 <= -2e-255)
		tmp = t_1;
	elseif (t_2 <= 5e-278)
		tmp = fma(fma(t, -1.0, x), Float64(Float64(y - a) / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$2, -2e-255], t$95$1, If[LessEqual[t$95$2, 5e-278], N[(N[(t * -1.0 + x), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2e-255 or 4.99999999999999985e-278 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 91.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 91.1

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. neg-sub0 N/A

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

      10. lift--.f64 N/A

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

      11. sub-neg N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. remove-double-neg N/A

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

      16. lower--.f64 N/A

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

      17. neg-sub0 N/A

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

      18. lift--.f64 N/A

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

      19. sub-neg N/A

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

      20. +-commutative N/A

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

      21. associate--r+ N/A

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

      22. neg-sub0 N/A

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

      23. remove-double-neg N/A

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

      24. lower--.f64 91.1

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

   4. Applied rewrites 91.1%

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

   if -2e-255 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 4.99999999999999985e-278

   1. Initial program 6.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 93.4%

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

4. Final simplification 91.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq -2 \cdot 10^{-255}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \mathbf{elif}\;\frac{t - x}{a - z} \cdot \left(y - z\right) + x \leq 5 \cdot 10^{-278}:\\ \;\;\;\;\mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z - a}, y - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\mathsf{fma}\left(t, -1, x\right), \frac{y - a}{z}, t\right)\\ \mathbf{if}\;z \leq -6.2 \cdot 10^{-81}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (fma t -1.0 x) (/ (- y a) z) t)))
   (if (<= z -6.2e-81)
     t_1
     (if (<= z 1.05e+33) (fma (/ (- y z) a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(fma(t, -1.0, x), ((y - a) / z), t);
	double tmp;
	if (z <= -6.2e-81) {
		tmp = t_1;
	} else if (z <= 1.05e+33) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(fma(t, -1.0, x), Float64(Float64(y - a) / z), t)
	tmp = 0.0
	if (z <= -6.2e-81)
		tmp = t_1;
	elseif (z <= 1.05e+33)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t * -1.0 + x), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -6.2e-81], t$95$1, If[LessEqual[z, 1.05e+33], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.19999999999999976e-81 or 1.05e33 < z

   1. Initial program 57.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 80.3%

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

   if -6.19999999999999976e-81 < z < 1.05e33

   1. Initial program 95.7%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 82.3

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

   5. Applied rewrites 82.3%

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

Alternative 5: 41.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -3.9 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-131}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{t}{a - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t) (/ y z) t)))
   (if (<= z -3.9e+112)
     t_1
     (if (<= z -1.8e-131)
       (* (/ (- x t) z) y)
       (if (<= z 2.5e+27) (* (/ t (- a z)) y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-t, (y / z), t);
	double tmp;
	if (z <= -3.9e+112) {
		tmp = t_1;
	} else if (z <= -1.8e-131) {
		tmp = ((x - t) / z) * y;
	} else if (z <= 2.5e+27) {
		tmp = (t / (a - z)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -3.9e+112)
		tmp = t_1;
	elseif (z <= -1.8e-131)
		tmp = Float64(Float64(Float64(x - t) / z) * y);
	elseif (z <= 2.5e+27)
		tmp = Float64(Float64(t / Float64(a - z)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -3.9e+112], t$95$1, If[LessEqual[z, -1.8e-131], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 2.5e+27], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.89999999999999968e112 or 2.4999999999999999e27 < z

   1. Initial program 53.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 26.2%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 77.0%

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

   12. Taylor expanded in t around inf

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

   14. Applied rewrites 59.8%

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

   if -3.89999999999999968e112 < z < -1.8e-131

   1. Initial program 79.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 37.5%

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

   if -1.8e-131 < z < 2.4999999999999999e27

   1. Initial program 95.8%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 41.9

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

   5. Applied rewrites 41.9%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 39.1%

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

   10. Applied rewrites 39.1%

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

4. Final simplification 46.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{elif}\;z \leq -1.8 \cdot 10^{-131}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;z \leq 2.5 \cdot 10^{+27}:\\ \;\;\;\;\frac{t}{a - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 71.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-81}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+33}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - z}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.2e-81)
   (fma (- x t) (/ y z) t)
   (if (<= z 1.05e+33) (fma (/ (- y z) a) (- t x) x) (fma (/ (- x t) z) y t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.2e-81) {
		tmp = fma((x - t), (y / z), t);
	} else if (z <= 1.05e+33) {
		tmp = fma(((y - z) / a), (t - x), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.2e-81)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	elseif (z <= 1.05e+33)
		tmp = fma(Float64(Float64(y - z) / a), Float64(t - x), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.2e-81], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 1.05e+33], N[(N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.19999999999999976e-81

   1. Initial program 63.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 35.1%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   if -6.19999999999999976e-81 < z < 1.05e33

   1. Initial program 95.7%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 82.3

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

   5. Applied rewrites 82.3%

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

   if 1.05e33 < z

   1. Initial program 51.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 80.1%

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

Alternative 7: 30.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) + x\\ \mathbf{if}\;z \leq -5.3 \cdot 10^{+112}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- t x) x)))
   (if (<= z -5.3e+112)
     t_1
     (if (<= z -2.1e-131)
       (* (/ x z) y)
       (if (<= z 3.1e+31) (* (/ 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 <= -5.3e+112) {
		tmp = t_1;
	} else if (z <= -2.1e-131) {
		tmp = (x / z) * y;
	} else if (z <= 3.1e+31) {
		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 <= (-5.3d+112)) then
        tmp = t_1
    else if (z <= (-2.1d-131)) then
        tmp = (x / z) * y
    else if (z <= 3.1d+31) 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 <= -5.3e+112) {
		tmp = t_1;
	} else if (z <= -2.1e-131) {
		tmp = (x / z) * y;
	} else if (z <= 3.1e+31) {
		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 <= -5.3e+112:
		tmp = t_1
	elif z <= -2.1e-131:
		tmp = (x / z) * y
	elif z <= 3.1e+31:
		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 <= -5.3e+112)
		tmp = t_1;
	elseif (z <= -2.1e-131)
		tmp = Float64(Float64(x / z) * y);
	elseif (z <= 3.1e+31)
		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 <= -5.3e+112)
		tmp = t_1;
	elseif (z <= -2.1e-131)
		tmp = (x / z) * y;
	elseif (z <= 3.1e+31)
		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, -5.3e+112], t$95$1, If[LessEqual[z, -2.1e-131], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 3.1e+31], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.30000000000000018e112 or 3.1000000000000002e31 < z

   1. Initial program 53.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 37.4

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

   5. Applied rewrites 37.4%

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

   if -5.30000000000000018e112 < z < -2.09999999999999997e-131

   1. Initial program 79.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 51.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 30.9%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 29.8%

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

   if -2.09999999999999997e-131 < z < 3.1000000000000002e31

   1. Initial program 95.8%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 42.2

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

   5. Applied rewrites 42.2%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 37.0%

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

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.3 \cdot 10^{+112}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{elif}\;z \leq -2.1 \cdot 10^{-131}:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-81}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{elif}\;z \leq 6.6 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.2e-81)
   (fma (- x t) (/ y z) t)
   (if (<= z 6.6e+56) (fma (- t x) (/ y a) x) (fma (/ (- x t) z) y t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.2e-81) {
		tmp = fma((x - t), (y / z), t);
	} else if (z <= 6.6e+56) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.2e-81)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	elseif (z <= 6.6e+56)
		tmp = fma(Float64(t - x), Float64(y / a), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.2e-81], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 6.6e+56], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.19999999999999976e-81

   1. Initial program 63.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 35.1%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 69.7%

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

   if -6.19999999999999976e-81 < z < 6.60000000000000004e56

   1. Initial program 95.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 25.8%

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

   7. Applied rewrites 25.8%

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

   8. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 81.5

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

   10. Applied rewrites 81.5%

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

   if 6.60000000000000004e56 < z

   1. Initial program 50.1%

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

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

   5. Applied rewrites 85.6%

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

   6. Taylor expanded in a around 0

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

   8. Applied rewrites 81.1%

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

Alternative 9: 65.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{t}{a} \cdot y + x\\ \mathbf{if}\;a \leq -7.4 \cdot 10^{+111}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (* (/ t a) y) x)))
   (if (<= a -7.4e+111) t_1 (if (<= a 0.0006) (fma (- x t) (/ y z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((t / a) * y) + x;
	double tmp;
	if (a <= -7.4e+111) {
		tmp = t_1;
	} else if (a <= 0.0006) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(t / a) * y) + x)
	tmp = 0.0
	if (a <= -7.4e+111)
		tmp = t_1;
	elseif (a <= 0.0006)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -7.4e+111], t$95$1, If[LessEqual[a, 0.0006], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.4000000000000005e111 or 5.99999999999999947e-4 < a

   1. Initial program 92.4%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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 9.8

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

   5. Applied rewrites 9.8%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 68.6

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

   8. Applied rewrites 68.6%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 72.7%

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

   if -7.4000000000000005e111 < a < 5.99999999999999947e-4

   1. Initial program 67.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 72.7%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 31.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 70.9%

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

4. Final simplification 71.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.4 \cdot 10^{+111}:\\ \;\;\;\;\frac{t}{a} \cdot y + x\\ \mathbf{elif}\;a \leq 0.0006:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{a} \cdot y + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 56.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -4.7 \cdot 10^{+86}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{t}{a} \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t) (/ y z) t)))
   (if (<= z -4.7e+86) t_1 (if (<= z 2.4e+57) (+ (* (/ t a) y) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-t, (y / z), t);
	double tmp;
	if (z <= -4.7e+86) {
		tmp = t_1;
	} else if (z <= 2.4e+57) {
		tmp = ((t / a) * y) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -4.7e+86)
		tmp = t_1;
	elseif (z <= 2.4e+57)
		tmp = Float64(Float64(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[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -4.7e+86], t$95$1, If[LessEqual[z, 2.4e+57], N[(N[(N[(t / a), $MachinePrecision] * y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.7000000000000002e86 or 2.40000000000000005e57 < z

   1. Initial program 53.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 86.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 28.5%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 80.3%

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

   12. Taylor expanded in t around inf

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

   14. Applied rewrites 60.5%

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

   if -4.7000000000000002e86 < z < 2.40000000000000005e57

   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 inf

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

      1. lower--.f64 7.0

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

   5. Applied rewrites 7.0%

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

   6. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 70.5

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

   8. Applied rewrites 70.5%

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

   9. Taylor expanded in t around inf

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

   11. Applied rewrites 58.7%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.7 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{elif}\;z \leq 2.4 \cdot 10^{+57}:\\ \;\;\;\;\frac{t}{a} \cdot y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 33.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{y - a}{z} \cdot x\\ \mathbf{if}\;x \leq -500000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-72}:\\ \;\;\;\;\frac{t}{a - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- y a) z) x)))
   (if (<= x -500000.0) t_1 (if (<= x 1.32e-72) (* (/ t (- a z)) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - a) / z) * x;
	double tmp;
	if (x <= -500000.0) {
		tmp = t_1;
	} else if (x <= 1.32e-72) {
		tmp = (t / (a - z)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = ((y - a) / z) * x
    if (x <= (-500000.0d0)) then
        tmp = t_1
    else if (x <= 1.32d-72) then
        tmp = (t / (a - z)) * y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - a) / z) * x;
	double tmp;
	if (x <= -500000.0) {
		tmp = t_1;
	} else if (x <= 1.32e-72) {
		tmp = (t / (a - z)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((y - a) / z) * x
	tmp = 0
	if x <= -500000.0:
		tmp = t_1
	elif x <= 1.32e-72:
		tmp = (t / (a - z)) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y - a) / z) * x)
	tmp = 0.0
	if (x <= -500000.0)
		tmp = t_1;
	elseif (x <= 1.32e-72)
		tmp = Float64(Float64(t / Float64(a - z)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((y - a) / z) * x;
	tmp = 0.0;
	if (x <= -500000.0)
		tmp = t_1;
	elseif (x <= 1.32e-72)
		tmp = (t / (a - z)) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[x, -500000.0], t$95$1, If[LessEqual[x, 1.32e-72], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -5e5 or 1.32000000000000001e-72 < x

   1. Initial program 68.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 37.1%

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

   if -5e5 < x < 1.32000000000000001e-72

   1. Initial program 86.8%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 41.6%

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

   10. Applied rewrites 42.4%

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

4. Final simplification 39.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -500000:\\ \;\;\;\;\frac{y - a}{z} \cdot x\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-72}:\\ \;\;\;\;\frac{t}{a - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y - a}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 31.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{x - t}{z} \cdot y\\ \mathbf{if}\;x \leq -980000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-72}:\\ \;\;\;\;\frac{t}{a - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* (/ (- x t) z) y)))
   (if (<= x -980000.0) t_1 (if (<= x 1.32e-72) (* (/ t (- a z)) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((x - t) / z) * y;
	double tmp;
	if (x <= -980000.0) {
		tmp = t_1;
	} else if (x <= 1.32e-72) {
		tmp = (t / (a - z)) * y;
	} 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 = ((x - t) / z) * y
    if (x <= (-980000.0d0)) then
        tmp = t_1
    else if (x <= 1.32d-72) then
        tmp = (t / (a - z)) * y
    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 = ((x - t) / z) * y;
	double tmp;
	if (x <= -980000.0) {
		tmp = t_1;
	} else if (x <= 1.32e-72) {
		tmp = (t / (a - z)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = ((x - t) / z) * y
	tmp = 0
	if x <= -980000.0:
		tmp = t_1
	elif x <= 1.32e-72:
		tmp = (t / (a - z)) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(x - t) / z) * y)
	tmp = 0.0
	if (x <= -980000.0)
		tmp = t_1;
	elseif (x <= 1.32e-72)
		tmp = Float64(Float64(t / Float64(a - z)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = ((x - t) / z) * y;
	tmp = 0.0;
	if (x <= -980000.0)
		tmp = t_1;
	elseif (x <= 1.32e-72)
		tmp = (t / (a - z)) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[x, -980000.0], t$95$1, If[LessEqual[x, 1.32e-72], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -9.8e5 or 1.32000000000000001e-72 < x

   1. Initial program 68.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 57.3%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 33.1%

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

   if -9.8e5 < x < 1.32000000000000001e-72

   1. Initial program 86.8%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 65.5

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

   5. Applied rewrites 65.5%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 41.6%

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

   10. Applied rewrites 42.4%

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

4. Final simplification 37.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -980000:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-72}:\\ \;\;\;\;\frac{t}{a - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{x - t}{z} \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-72}:\\ \;\;\;\;\frac{t}{a - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -19.0)
   (* (/ x z) y)
   (if (<= x 1.32e-72) (* (/ t (- a z)) y) (* (/ y z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -19.0) {
		tmp = (x / z) * y;
	} else if (x <= 1.32e-72) {
		tmp = (t / (a - z)) * y;
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-19.0d0)) then
        tmp = (x / z) * y
    else if (x <= 1.32d-72) then
        tmp = (t / (a - z)) * y
    else
        tmp = (y / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -19.0) {
		tmp = (x / z) * y;
	} else if (x <= 1.32e-72) {
		tmp = (t / (a - z)) * y;
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -19.0:
		tmp = (x / z) * y
	elif x <= 1.32e-72:
		tmp = (t / (a - z)) * y
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -19.0)
		tmp = Float64(Float64(x / z) * y);
	elseif (x <= 1.32e-72)
		tmp = Float64(Float64(t / Float64(a - z)) * y);
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -19.0)
		tmp = (x / z) * y;
	elseif (x <= 1.32e-72)
		tmp = (t / (a - z)) * y;
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -19.0], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 1.32e-72], N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -19

   1. Initial program 60.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 39.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 31.4%

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

   if -19 < x < 1.32000000000000001e-72

   1. Initial program 86.7%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 66.0

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 41.9%

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

   10. Applied rewrites 42.7%

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

   if 1.32000000000000001e-72 < x

   1. Initial program 75.1%

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

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 34.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 32.7%

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

4. Final simplification 36.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -19:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-72}:\\ \;\;\;\;\frac{t}{a - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 30.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -8.5:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-72}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -8.5)
   (* (/ x z) y)
   (if (<= x 1.32e-72) (* (/ y (- a z)) t) (* (/ y z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -8.5) {
		tmp = (x / z) * y;
	} else if (x <= 1.32e-72) {
		tmp = (y / (a - z)) * t;
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-8.5d0)) then
        tmp = (x / z) * y
    else if (x <= 1.32d-72) then
        tmp = (y / (a - z)) * t
    else
        tmp = (y / z) * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -8.5) {
		tmp = (x / z) * y;
	} else if (x <= 1.32e-72) {
		tmp = (y / (a - z)) * t;
	} else {
		tmp = (y / z) * x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -8.5:
		tmp = (x / z) * y
	elif x <= 1.32e-72:
		tmp = (y / (a - z)) * t
	else:
		tmp = (y / z) * x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -8.5)
		tmp = Float64(Float64(x / z) * y);
	elseif (x <= 1.32e-72)
		tmp = Float64(Float64(y / Float64(a - z)) * t);
	else
		tmp = Float64(Float64(y / z) * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -8.5)
		tmp = (x / z) * y;
	elseif (x <= 1.32e-72)
		tmp = (y / (a - z)) * t;
	else
		tmp = (y / z) * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -8.5], N[(N[(x / z), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[x, 1.32e-72], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision], N[(N[(y / z), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -8.5

   1. Initial program 60.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 58.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 39.6%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 31.4%

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

   if -8.5 < x < 1.32000000000000001e-72

   1. Initial program 86.7%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 66.0

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

   5. Applied rewrites 66.0%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 41.9%

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

   if 1.32000000000000001e-72 < x

   1. Initial program 75.1%

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

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

   5. Applied rewrites 56.0%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 34.8%

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

   9. Taylor expanded in a around 0

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

   11. Applied rewrites 32.7%

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

4. Final simplification 36.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -8.5:\\ \;\;\;\;\frac{x}{z} \cdot y\\ \mathbf{elif}\;x \leq 1.32 \cdot 10^{-72}:\\ \;\;\;\;\frac{y}{a - z} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\frac{y}{z} \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 30.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) + x\\ \mathbf{if}\;z \leq -2.3 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- t x) x)))
   (if (<= z -2.3e+96) t_1 (if (<= z 3.1e+31) (* (/ 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 <= -2.3e+96) {
		tmp = t_1;
	} else if (z <= 3.1e+31) {
		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 <= (-2.3d+96)) then
        tmp = t_1
    else if (z <= 3.1d+31) 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 <= -2.3e+96) {
		tmp = t_1;
	} else if (z <= 3.1e+31) {
		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 <= -2.3e+96:
		tmp = t_1
	elif z <= 3.1e+31:
		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 <= -2.3e+96)
		tmp = t_1;
	elseif (z <= 3.1e+31)
		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 <= -2.3e+96)
		tmp = t_1;
	elseif (z <= 3.1e+31)
		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, -2.3e+96], t$95$1, If[LessEqual[z, 3.1e+31], N[(N[(y / a), $MachinePrecision] * t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.30000000000000015e96 or 3.1000000000000002e31 < z

   1. Initial program 53.9%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 37.3

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

   5. Applied rewrites 37.3%

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

   if -2.30000000000000015e96 < z < 3.1000000000000002e31

   1. Initial program 91.3%

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

   3. Taylor expanded in t around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 38.9

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

   5. Applied rewrites 38.9%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 30.9%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.3 \cdot 10^{+96}:\\ \;\;\;\;\left(t - x\right) + x\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+31}:\\ \;\;\;\;\frac{y}{a} \cdot t\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 19.0% 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 76.5%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
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 18.7

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

5. Applied rewrites 18.7%

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

6. Final simplification 18.7%

   \[\leadsto \left(t - x\right) + x \]
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 76.5%

   \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
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 18.7

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

5. Applied rewrites 18.7%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 2.9%

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

9. Final simplification 2.9%

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

Reproduce

herbie shell --seed 2024270 
(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)))))