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

Percentage Accurate: 67.8% → 89.8%

Time: 10.9s

Alternatives: 18

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 67.8% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 89.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-237}:\\ \;\;\;\;x - \frac{x - y}{\frac{1}{t - z} \cdot \left(t - a\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-283}:\\ \;\;\;\;y - \frac{\mathsf{fma}\left(a, \frac{\left(a - z\right) \cdot \left(x - y\right)}{t}, z \cdot \left(y - x\right)\right) - a \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - y}{\frac{t - a}{t - z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* (- t z) (- y x)) (- a t)))))
   (if (<= t_1 -5e-237)
     (- x (/ (- x y) (* (/ 1.0 (- t z)) (- t a))))
     (if (<= t_1 2e-283)
       (-
        y
        (/
         (- (fma a (/ (* (- a z) (- x y)) t) (* z (- y x))) (* a (- y x)))
         t))
       (- x (/ (- x y) (/ (- t a) (- t z))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (((t - z) * (y - x)) / (a - t));
	double tmp;
	if (t_1 <= -5e-237) {
		tmp = x - ((x - y) / ((1.0 / (t - z)) * (t - a)));
	} else if (t_1 <= 2e-283) {
		tmp = y - ((fma(a, (((a - z) * (x - y)) / t), (z * (y - x))) - (a * (y - x))) / t);
	} else {
		tmp = x - ((x - y) / ((t - a) / (t - z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(Float64(t - z) * Float64(y - x)) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -5e-237)
		tmp = Float64(x - Float64(Float64(x - y) / Float64(Float64(1.0 / Float64(t - z)) * Float64(t - a))));
	elseif (t_1 <= 2e-283)
		tmp = Float64(y - Float64(Float64(fma(a, Float64(Float64(Float64(a - z) * Float64(x - y)) / t), Float64(z * Float64(y - x))) - Float64(a * Float64(y - x))) / t));
	else
		tmp = Float64(x - Float64(Float64(x - y) / Float64(Float64(t - a) / Float64(t - z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(N[(t - z), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-237], N[(x - N[(N[(x - y), $MachinePrecision] / N[(N[(1.0 / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-283], N[(y - N[(N[(N[(a * N[(N[(N[(a - z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(z * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - N[(a * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x - y), $MachinePrecision] / N[(N[(t - a), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -5.0000000000000002e-237

   1. Initial program 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval 89.1

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

   6. Applied rewrites 89.1%

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

   if -5.0000000000000002e-237 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 1.99999999999999989e-283

   1. Initial program 5.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 6.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 6.2

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

   4. Applied rewrites 6.2%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 3.7

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

   7. Applied rewrites 3.7%

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

   8. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   10. Applied rewrites 99.8%

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

   if 1.99999999999999989e-283 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 90.2

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

   4. Applied rewrites 90.2%

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

4. Final simplification 90.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq -5 \cdot 10^{-237}:\\ \;\;\;\;x - \frac{x - y}{\frac{1}{t - z} \cdot \left(t - a\right)}\\ \mathbf{elif}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq 2 \cdot 10^{-283}:\\ \;\;\;\;y - \frac{\mathsf{fma}\left(a, \frac{\left(a - z\right) \cdot \left(x - y\right)}{t}, z \cdot \left(y - x\right)\right) - a \cdot \left(y - x\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - y}{\frac{t - a}{t - z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ t_2 := x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t}, \left(t - z\right) \cdot \left(x - y\right), x\right)\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x y) t) (- z a) y))
        (t_2 (- x (/ (* (- t z) (- y x)) (- a t)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -5e-237)
       (fma (/ 1.0 (- a t)) (* (- t z) (- x y)) x)
       (if (<= t_2 0.0)
         (fma (/ x t) (- z a) y)
         (if (<= t_2 5e+303) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - y) / t), (z - a), y);
	double t_2 = x - (((t - z) * (y - x)) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -5e-237) {
		tmp = fma((1.0 / (a - t)), ((t - z) * (x - y)), x);
	} else if (t_2 <= 0.0) {
		tmp = fma((x / t), (z - a), y);
	} else if (t_2 <= 5e+303) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - y) / t), Float64(z - a), y)
	t_2 = Float64(x - Float64(Float64(Float64(t - z) * Float64(y - x)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -5e-237)
		tmp = fma(Float64(1.0 / Float64(a - t)), Float64(Float64(t - z) * Float64(x - y)), x);
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(x / t), Float64(z - a), y);
	elseif (t_2 <= 5e+303)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(N[(t - z), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -5e-237], N[(N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(N[(t - z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$2, 0.0], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t$95$2, 5e+303], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 4 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -inf.0 or 4.9999999999999997e303 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 31.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 71.3%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -5.0000000000000002e-237

   1. Initial program 95.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 95.4

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 95.4

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

   4. Applied rewrites 95.4%

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

   if -5.0000000000000002e-237 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.0%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 88.0%

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

   if 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 4.9999999999999997e303

   1. Initial program 96.7%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{elif}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq -5 \cdot 10^{-237}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a - t}, \left(t - z\right) \cdot \left(x - y\right), x\right)\\ \mathbf{elif}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{elif}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 85.7% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ t_2 := x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -\infty:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -5 \cdot 10^{-237}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+303}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x y) t) (- z a) y))
        (t_2 (- x (/ (* (- t z) (- y x)) (- a t)))))
   (if (<= t_2 (- INFINITY))
     t_1
     (if (<= t_2 -5e-237)
       t_2
       (if (<= t_2 0.0)
         (fma (/ x t) (- z a) y)
         (if (<= t_2 5e+303) t_2 t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - y) / t), (z - a), y);
	double t_2 = x - (((t - z) * (y - x)) / (a - t));
	double tmp;
	if (t_2 <= -((double) INFINITY)) {
		tmp = t_1;
	} else if (t_2 <= -5e-237) {
		tmp = t_2;
	} else if (t_2 <= 0.0) {
		tmp = fma((x / t), (z - a), y);
	} else if (t_2 <= 5e+303) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - y) / t), Float64(z - a), y)
	t_2 = Float64(x - Float64(Float64(Float64(t - z) * Float64(y - x)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= Float64(-Inf))
		tmp = t_1;
	elseif (t_2 <= -5e-237)
		tmp = t_2;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(x / t), Float64(z - a), y);
	elseif (t_2 <= 5e+303)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(N[(t - z), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, (-Infinity)], t$95$1, If[LessEqual[t$95$2, -5e-237], t$95$2, If[LessEqual[t$95$2, 0.0], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t$95$2, 5e+303], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -inf.0 or 4.9999999999999997e303 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 31.4%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 71.3%

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

   if -inf.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -5.0000000000000002e-237 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 4.9999999999999997e303

   1. Initial program 96.0%

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

   if -5.0000000000000002e-237 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.0%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 88.0%

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

4. Final simplification 85.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq -\infty:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{elif}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq -5 \cdot 10^{-237}:\\ \;\;\;\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t}\\ \mathbf{elif}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{elif}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq 5 \cdot 10^{+303}:\\ \;\;\;\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 89.5% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-237}:\\ \;\;\;\;x - \frac{x - y}{\frac{1}{t - z} \cdot \left(t - a\right)}\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-283}:\\ \;\;\;\;y - \frac{\mathsf{fma}\left(\frac{y - x}{t} \cdot \left(z - a\right), a, \left(a - z\right) \cdot \left(x - y\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - y}{\frac{t - a}{t - z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* (- t z) (- y x)) (- a t)))))
   (if (<= t_1 -5e-237)
     (- x (/ (- x y) (* (/ 1.0 (- t z)) (- t a))))
     (if (<= t_1 2e-283)
       (- y (/ (fma (* (/ (- y x) t) (- z a)) a (* (- a z) (- x y))) t))
       (- x (/ (- x y) (/ (- t a) (- t z))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (((t - z) * (y - x)) / (a - t));
	double tmp;
	if (t_1 <= -5e-237) {
		tmp = x - ((x - y) / ((1.0 / (t - z)) * (t - a)));
	} else if (t_1 <= 2e-283) {
		tmp = y - (fma((((y - x) / t) * (z - a)), a, ((a - z) * (x - y))) / t);
	} else {
		tmp = x - ((x - y) / ((t - a) / (t - z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(Float64(t - z) * Float64(y - x)) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -5e-237)
		tmp = Float64(x - Float64(Float64(x - y) / Float64(Float64(1.0 / Float64(t - z)) * Float64(t - a))));
	elseif (t_1 <= 2e-283)
		tmp = Float64(y - Float64(fma(Float64(Float64(Float64(y - x) / t) * Float64(z - a)), a, Float64(Float64(a - z) * Float64(x - y))) / t));
	else
		tmp = Float64(x - Float64(Float64(x - y) / Float64(Float64(t - a) / Float64(t - z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(N[(t - z), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-237], N[(x - N[(N[(x - y), $MachinePrecision] / N[(N[(1.0 / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 2e-283], N[(y - N[(N[(N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision]), $MachinePrecision] * a + N[(N[(a - z), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], N[(x - N[(N[(x - y), $MachinePrecision] / N[(N[(t - a), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -5.0000000000000002e-237

   1. Initial program 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval 89.1

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

   6. Applied rewrites 89.1%

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

   if -5.0000000000000002e-237 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 1.99999999999999989e-283

   1. Initial program 5.8%

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

   3. Taylor expanded in t around -inf

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

   5. Applied rewrites 93.4%

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

   if 1.99999999999999989e-283 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 72.7%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 90.2

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

   4. Applied rewrites 90.2%

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

4. Final simplification 89.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq -5 \cdot 10^{-237}:\\ \;\;\;\;x - \frac{x - y}{\frac{1}{t - z} \cdot \left(t - a\right)}\\ \mathbf{elif}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq 2 \cdot 10^{-283}:\\ \;\;\;\;y - \frac{\mathsf{fma}\left(\frac{y - x}{t} \cdot \left(z - a\right), a, \left(a - z\right) \cdot \left(x - y\right)\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - y}{\frac{t - a}{t - z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-237}:\\ \;\;\;\;x - \frac{x - y}{\frac{1}{t - z} \cdot \left(t - a\right)}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - y}{\frac{t - a}{t - z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (* (- t z) (- y x)) (- a t)))))
   (if (<= t_1 -5e-237)
     (- x (/ (- x y) (* (/ 1.0 (- t z)) (- t a))))
     (if (<= t_1 0.0)
       (fma (/ x t) (- z a) y)
       (- x (/ (- x y) (/ (- t a) (- t z))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (((t - z) * (y - x)) / (a - t));
	double tmp;
	if (t_1 <= -5e-237) {
		tmp = x - ((x - y) / ((1.0 / (t - z)) * (t - a)));
	} else if (t_1 <= 0.0) {
		tmp = fma((x / t), (z - a), y);
	} else {
		tmp = x - ((x - y) / ((t - a) / (t - z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(Float64(t - z) * Float64(y - x)) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -5e-237)
		tmp = Float64(x - Float64(Float64(x - y) / Float64(Float64(1.0 / Float64(t - z)) * Float64(t - a))));
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(x / t), Float64(z - a), y);
	else
		tmp = Float64(x - Float64(Float64(x - y) / Float64(Float64(t - a) / Float64(t - z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(N[(t - z), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-237], N[(x - N[(N[(x - y), $MachinePrecision] / N[(N[(1.0 / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(t - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], N[(x - N[(N[(x - y), $MachinePrecision] / N[(N[(t - a), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -5.0000000000000002e-237

   1. Initial program 65.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.1

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

   4. Applied rewrites 89.1%

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

      1. lift-/.f64 N/A

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

      2. div-inv N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval 89.1

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

   6. Applied rewrites 89.1%

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

   if -5.0000000000000002e-237 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.0%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 88.0%

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

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

   1. Initial program 72.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.7

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

   4. Applied rewrites 89.7%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq -5 \cdot 10^{-237}:\\ \;\;\;\;x - \frac{x - y}{\frac{1}{t - z} \cdot \left(t - a\right)}\\ \mathbf{elif}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - y}{\frac{t - a}{t - z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{x - y}{\frac{t - a}{t - z}}\\ t_2 := x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-237}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (/ (- x y) (/ (- t a) (- t z)))))
        (t_2 (- x (/ (* (- t z) (- y x)) (- a t)))))
   (if (<= t_2 -5e-237) t_1 (if (<= t_2 0.0) (fma (/ x t) (- z a) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((x - y) / ((t - a) / (t - z)));
	double t_2 = x - (((t - z) * (y - x)) / (a - t));
	double tmp;
	if (t_2 <= -5e-237) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma((x / t), (z - a), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(x - y) / Float64(Float64(t - a) / Float64(t - z))))
	t_2 = Float64(x - Float64(Float64(Float64(t - z) * Float64(y - x)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -5e-237)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(x / t), Float64(z - a), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(x - y), $MachinePrecision] / N[(N[(t - a), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(x - N[(N[(N[(t - z), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-237], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -5.0000000000000002e-237 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 68.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. clear-num N/A

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

      5. un-div-inv N/A

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

      6. lower-/.f64 N/A

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

      7. lower-/.f64 89.4

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

   4. Applied rewrites 89.4%

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

   if -5.0000000000000002e-237 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.0%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 88.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 88.0%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq -5 \cdot 10^{-237}:\\ \;\;\;\;x - \frac{x - y}{\frac{t - a}{t - z}}\\ \mathbf{elif}\;x - \frac{\left(t - z\right) \cdot \left(y - x\right)}{a - t} \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{else}:\\ \;\;\;\;x - \frac{x - y}{\frac{t - a}{t - z}}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 48.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -3 \cdot 10^{+25}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -7.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{z}{t - a} \cdot x\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-110}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-43}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y a) x)))
   (if (<= a -3e+25)
     t_1
     (if (<= a -7.2e-175)
       (* (/ z (- t a)) x)
       (if (<= a 1.7e-110)
         (* (/ (- x y) t) z)
         (if (<= a 3.2e-43) (+ (- y x) x) t_1))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / a), x);
	double tmp;
	if (a <= -3e+25) {
		tmp = t_1;
	} else if (a <= -7.2e-175) {
		tmp = (z / (t - a)) * x;
	} else if (a <= 1.7e-110) {
		tmp = ((x - y) / t) * z;
	} else if (a <= 3.2e-43) {
		tmp = (y - x) + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / a), x)
	tmp = 0.0
	if (a <= -3e+25)
		tmp = t_1;
	elseif (a <= -7.2e-175)
		tmp = Float64(Float64(z / Float64(t - a)) * x);
	elseif (a <= 1.7e-110)
		tmp = Float64(Float64(Float64(x - y) / t) * z);
	elseif (a <= 3.2e-43)
		tmp = Float64(Float64(y - x) + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -3e+25], t$95$1, If[LessEqual[a, -7.2e-175], N[(N[(z / N[(t - a), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[a, 1.7e-110], N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[a, 3.2e-43], N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if a < -3.00000000000000006e25 or 3.19999999999999985e-43 < a

   1. Initial program 65.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 65.7

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 65.7

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

   4. Applied rewrites 65.7%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 65.5

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

   7. Applied rewrites 65.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 59.0%

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

   if -3.00000000000000006e25 < a < -7.2e-175

   1. Initial program 63.1%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 42.8%

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

   if -7.2e-175 < a < 1.7000000000000001e-110

   1. Initial program 59.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 86.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 51.9%

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

   if 1.7000000000000001e-110 < a < 3.19999999999999985e-43

   1. Initial program 71.6%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 46.5

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

   5. Applied rewrites 46.5%

      \[\leadsto x + \color{blue}{\left(y - x\right)} \]
3. Recombined 4 regimes into one program.

4. Final simplification 54.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -3 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;a \leq -7.2 \cdot 10^{-175}:\\ \;\;\;\;\frac{z}{t - a} \cdot x\\ \mathbf{elif}\;a \leq 1.7 \cdot 10^{-110}:\\ \;\;\;\;\frac{x - y}{t} \cdot z\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-43}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 58.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -4.7 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t \leq 2.6 \cdot 10^{-57}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x t) (- z a) y)))
   (if (<= t -4.7e+96)
     t_1
     (if (<= t -4.5e-170)
       (fma z (/ y a) x)
       (if (<= t 2.6e-57) (fma z (/ (- x) a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x / t), (z - a), y);
	double tmp;
	if (t <= -4.7e+96) {
		tmp = t_1;
	} else if (t <= -4.5e-170) {
		tmp = fma(z, (y / a), x);
	} else if (t <= 2.6e-57) {
		tmp = fma(z, (-x / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -4.7e+96)
		tmp = t_1;
	elseif (t <= -4.5e-170)
		tmp = fma(z, Float64(y / a), x);
	elseif (t <= 2.6e-57)
		tmp = fma(z, Float64(Float64(-x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -4.7e+96], t$95$1, If[LessEqual[t, -4.5e-170], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 2.6e-57], N[(z * N[((-x) / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.7000000000000001e96 or 2.59999999999999985e-57 < t

   1. Initial program 37.0%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 71.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 68.3%

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

   if -4.7000000000000001e96 < t < -4.50000000000000002e-170

   1. Initial program 75.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 75.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 58.2

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 56.5%

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

   if -4.50000000000000002e-170 < t < 2.59999999999999985e-57

   1. Initial program 94.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 94.2

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 94.2

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

   4. Applied rewrites 94.2%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 91.9

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

   7. Applied rewrites 91.9%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(z, \frac{-1 \cdot x}{a}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 77.3%

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

Alternative 9: 43.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) + x\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- y x) x)))
   (if (<= t -1.25e+102)
     t_1
     (if (<= t -4.5e-170)
       (fma z (/ y a) x)
       (if (<= t 1.85e+200) (fma z (/ (- x) a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -1.25e+102) {
		tmp = t_1;
	} else if (t <= -4.5e-170) {
		tmp = fma(z, (y / a), x);
	} else if (t <= 1.85e+200) {
		tmp = fma(z, (-x / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) + x)
	tmp = 0.0
	if (t <= -1.25e+102)
		tmp = t_1;
	elseif (t <= -4.5e-170)
		tmp = fma(z, Float64(y / a), x);
	elseif (t <= 1.85e+200)
		tmp = fma(z, Float64(Float64(-x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.25e+102], t$95$1, If[LessEqual[t, -4.5e-170], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.85e+200], N[(z * N[((-x) / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.25e102 or 1.8500000000000001e200 < t

   1. Initial program 25.9%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 41.7

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

   5. Applied rewrites 41.7%

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

   if -1.25e102 < t < -4.50000000000000002e-170

   1. Initial program 75.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 75.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 58.2

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 56.5%

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

   if -4.50000000000000002e-170 < t < 1.8500000000000001e200

   1. Initial program 81.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 82.1

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 82.1

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in y around 0

      \[\leadsto \mathsf{fma}\left(z, \frac{-1 \cdot x}{a}, x\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 62.3%

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

4. Final simplification 55.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+102}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{elif}\;t \leq -4.5 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{-x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 42.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) + x\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+200}:\\ \;\;\;\;x - \frac{z \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- y x) x)))
   (if (<= t -1.25e+102)
     t_1
     (if (<= t -4.8e-170)
       (fma z (/ y a) x)
       (if (<= t 1.85e+200) (- x (/ (* z x) a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -1.25e+102) {
		tmp = t_1;
	} else if (t <= -4.8e-170) {
		tmp = fma(z, (y / a), x);
	} else if (t <= 1.85e+200) {
		tmp = x - ((z * x) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) + x)
	tmp = 0.0
	if (t <= -1.25e+102)
		tmp = t_1;
	elseif (t <= -4.8e-170)
		tmp = fma(z, Float64(y / a), x);
	elseif (t <= 1.85e+200)
		tmp = Float64(x - Float64(Float64(z * x) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.25e+102], t$95$1, If[LessEqual[t, -4.8e-170], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 1.85e+200], N[(x - N[(N[(z * x), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.25e102 or 1.8500000000000001e200 < t

   1. Initial program 25.9%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 41.7

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

   5. Applied rewrites 41.7%

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

   if -1.25e102 < t < -4.7999999999999999e-170

   1. Initial program 75.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 75.0

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 75.0

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

   4. Applied rewrites 75.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 58.2

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

   7. Applied rewrites 58.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 56.5%

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

   if -4.7999999999999999e-170 < t < 1.8500000000000001e200

   1. Initial program 81.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 82.1

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 82.1

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

   4. Applied rewrites 82.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.8

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

   7. Applied rewrites 73.8%

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

   8. Taylor expanded in y around 0

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

   10. Applied rewrites 59.9%

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

4. Final simplification 53.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+102}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{elif}\;t \leq -4.8 \cdot 10^{-170}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+200}:\\ \;\;\;\;x - \frac{z \cdot x}{a}\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 74.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -2.7 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+39}:\\ \;\;\;\;\mathsf{fma}\left(y - x, \frac{z - t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x y) t) (- z a) y)))
   (if (<= t -2.7e-64)
     t_1
     (if (<= t 4.1e+39) (fma (- y x) (/ (- z t) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - y) / t), (z - a), y);
	double tmp;
	if (t <= -2.7e-64) {
		tmp = t_1;
	} else if (t <= 4.1e+39) {
		tmp = fma((y - x), ((z - t) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - y) / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -2.7e-64)
		tmp = t_1;
	elseif (t <= 4.1e+39)
		tmp = fma(Float64(y - x), Float64(Float64(z - t) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.7e-64], t$95$1, If[LessEqual[t, 4.1e+39], N[(N[(y - x), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.69999999999999986e-64 or 4.10000000000000004e39 < t

   1. Initial program 42.3%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 69.0%

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

   if -2.69999999999999986e-64 < t < 4.10000000000000004e39

   1. Initial program 89.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 89.9

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 89.9

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

   4. Applied rewrites 89.9%

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

   5. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 88.5

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

   7. Applied rewrites 88.5%

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

Alternative 12: 72.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - y}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x y) t) (- z a) y)))
   (if (<= t -2.1e-64) t_1 (if (<= t 5.2e+35) (fma (/ (- y x) a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - y) / t), (z - a), y);
	double tmp;
	if (t <= -2.1e-64) {
		tmp = t_1;
	} else if (t <= 5.2e+35) {
		tmp = fma(((y - x) / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - y) / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -2.1e-64)
		tmp = t_1;
	elseif (t <= 5.2e+35)
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - y), $MachinePrecision] / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -2.1e-64], t$95$1, If[LessEqual[t, 5.2e+35], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.10000000000000011e-64 or 5.20000000000000013e35 < t

   1. Initial program 42.3%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 69.0%

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

   if -2.10000000000000011e-64 < t < 5.20000000000000013e35

   1. Initial program 89.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 83.8

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

   5. Applied rewrites 83.8%

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

Alternative 13: 68.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{t - z}{t - a} \cdot y\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -2.7e-64)
   (* (/ (- t z) (- t a)) y)
   (if (<= t 7.5e+36) (fma (/ (- y x) a) z x) (fma (/ x t) (- z a) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -2.7e-64) {
		tmp = ((t - z) / (t - a)) * y;
	} else if (t <= 7.5e+36) {
		tmp = fma(((y - x) / a), z, x);
	} else {
		tmp = fma((x / t), (z - a), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -2.7e-64)
		tmp = Float64(Float64(Float64(t - z) / Float64(t - a)) * y);
	elseif (t <= 7.5e+36)
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	else
		tmp = fma(Float64(x / t), Float64(z - a), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -2.7e-64], N[(N[(N[(t - z), $MachinePrecision] / N[(t - a), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[t, 7.5e+36], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -2.69999999999999986e-64

   1. Initial program 47.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 63.0%

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

   if -2.69999999999999986e-64 < t < 7.50000000000000054e36

   1. Initial program 89.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 83.8

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

   5. Applied rewrites 83.8%

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

   if 7.50000000000000054e36 < t

   1. Initial program 31.3%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 75.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 71.5%

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

4. Final simplification 74.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.7 \cdot 10^{-64}:\\ \;\;\;\;\frac{t - z}{t - a} \cdot y\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{t}, z - a, y\right)\\ \mathbf{if}\;t \leq -4.7 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7.5 \cdot 10^{+36}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - x}{a}, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x t) (- z a) y)))
   (if (<= t -4.7e+96) t_1 (if (<= t 7.5e+36) (fma (/ (- y x) a) z x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x / t), (z - a), y);
	double tmp;
	if (t <= -4.7e+96) {
		tmp = t_1;
	} else if (t <= 7.5e+36) {
		tmp = fma(((y - x) / a), z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x / t), Float64(z - a), y)
	tmp = 0.0
	if (t <= -4.7e+96)
		tmp = t_1;
	elseif (t <= 7.5e+36)
		tmp = fma(Float64(Float64(y - x) / a), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / t), $MachinePrecision] * N[(z - a), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -4.7e+96], t$95$1, If[LessEqual[t, 7.5e+36], N[(N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.7000000000000001e96 or 7.50000000000000054e36 < t

   1. Initial program 29.7%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. lower-fma.f64 N/A

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

   5. Applied rewrites 76.3%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 72.7%

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

   if -4.7000000000000001e96 < t < 7.50000000000000054e36

   1. Initial program 85.1%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 74.5

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

   5. Applied rewrites 74.5%

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

Alternative 15: 46.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) + x\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- y x) x)))
   (if (<= t -1.25e+102) t_1 (if (<= t 1.85e+200) (fma z (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -1.25e+102) {
		tmp = t_1;
	} else if (t <= 1.85e+200) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) + x)
	tmp = 0.0
	if (t <= -1.25e+102)
		tmp = t_1;
	elseif (t <= 1.85e+200)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -1.25e+102], t$95$1, If[LessEqual[t, 1.85e+200], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.25e102 or 1.8500000000000001e200 < t

   1. Initial program 25.9%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 41.7

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

   5. Applied rewrites 41.7%

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

   if -1.25e102 < t < 1.8500000000000001e200

   1. Initial program 79.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. clear-num N/A

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

      5. associate-/r/ N/A

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

      6. lower-fma.f64 N/A

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

      7. frac-2neg N/A

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

      8. metadata-eval N/A

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

      9. lower-/.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{0 - \left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a - t\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(a + \left(\mathsf{neg}\left(t\right)\right)\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{0 - \color{blue}{\left(\left(\mathsf{neg}\left(t\right)\right) + a\right)}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\frac{-1}{\color{blue}{\left(0 - \left(\mathsf{neg}\left(t\right)\right)\right) - a}}, \left(y - x\right) \cdot \left(z - t\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 79.6

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

      18. lift-*.f64 N/A

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

      19. *-commutative N/A

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

      20. lower-*.f64 79.6

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

   4. Applied rewrites 79.6%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 68.4

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

   7. Applied rewrites 68.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 53.8%

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

4. Final simplification 50.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.25 \cdot 10^{+102}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{elif}\;t \leq 1.85 \cdot 10^{+200}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 29.3% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - x\right) + x\\ \mathbf{if}\;t \leq -5.1 \cdot 10^{+96}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ (- y x) x)))
   (if (<= t -5.1e+96) t_1 (if (<= t 4.2e+89) (* (/ z a) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (y - x) + x;
	double tmp;
	if (t <= -5.1e+96) {
		tmp = t_1;
	} else if (t <= 4.2e+89) {
		tmp = (z / a) * 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 - x) + x
    if (t <= (-5.1d+96)) then
        tmp = t_1
    else if (t <= 4.2d+89) then
        tmp = (z / a) * 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 - x) + x;
	double tmp;
	if (t <= -5.1e+96) {
		tmp = t_1;
	} else if (t <= 4.2e+89) {
		tmp = (z / a) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (y - x) + x
	tmp = 0
	if t <= -5.1e+96:
		tmp = t_1
	elif t <= 4.2e+89:
		tmp = (z / a) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(y - x) + x)
	tmp = 0.0
	if (t <= -5.1e+96)
		tmp = t_1;
	elseif (t <= 4.2e+89)
		tmp = Float64(Float64(z / a) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (y - x) + x;
	tmp = 0.0;
	if (t <= -5.1e+96)
		tmp = t_1;
	elseif (t <= 4.2e+89)
		tmp = (z / a) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -5.1e+96], t$95$1, If[LessEqual[t, 4.2e+89], N[(N[(z / a), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -5.10000000000000015e96 or 4.19999999999999972e89 < t

   1. Initial program 29.5%

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

   3. Taylor expanded in t around inf

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

      1. lower--.f64 37.4

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

   5. Applied rewrites 37.4%

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

   if -5.10000000000000015e96 < t < 4.19999999999999972e89

   1. Initial program 84.2%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 35.5

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

   5. Applied rewrites 35.5%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 26.1%

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

   9. Taylor expanded in t around 0

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

   11. Applied rewrites 24.9%

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

4. Final simplification 29.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -5.1 \cdot 10^{+96}:\\ \;\;\;\;\left(y - x\right) + x\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+89}:\\ \;\;\;\;\frac{z}{a} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\left(y - x\right) + x\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 19.4% accurate, 4.1× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t, double a) {
	return (y - 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 = (y - x) + x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 64.5%

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

3. Taylor expanded in t around inf

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

   1. lower--.f64 19.0

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

5. Applied rewrites 19.0%

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

6. Final simplification 19.0%

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

Alternative 18: 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 64.5%

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

3. Taylor expanded in t around inf

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

   1. lower--.f64 19.0

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

5. Applied rewrites 19.0%

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

6. Taylor expanded in y around 0

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

8. Applied rewrites 2.7%

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

9. Final simplification 2.7%

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

Developer Target 1: 86.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{y - x}{1} \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;a < -1.6153062845442575 \cdot 10^{-142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a < 3.774403170083174 \cdot 10^{-182}:\\ \;\;\;\;y - \frac{z}{t} \cdot \left(y - x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (/ (- y x) 1.0) (/ (- z t) (- a t))))))
   (if (< a -1.6153062845442575e-142)
     t_1
     (if (< a 3.774403170083174e-182) (- y (* (/ z t) (- y x))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (((y - x) / 1.0d0) * ((z - t) / (a - t)))
    if (a < (-1.6153062845442575d-142)) then
        tmp = t_1
    else if (a < 3.774403170083174d-182) then
        tmp = y - ((z / t) * (y - x))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	double tmp;
	if (a < -1.6153062845442575e-142) {
		tmp = t_1;
	} else if (a < 3.774403170083174e-182) {
		tmp = y - ((z / t) * (y - x));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)))
	tmp = 0
	if a < -1.6153062845442575e-142:
		tmp = t_1
	elif a < 3.774403170083174e-182:
		tmp = y - ((z / t) * (y - x))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) / 1.0) * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = Float64(y - Float64(Float64(z / t) * Float64(y - x)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (((y - x) / 1.0) * ((z - t) / (a - t)));
	tmp = 0.0;
	if (a < -1.6153062845442575e-142)
		tmp = t_1;
	elseif (a < 3.774403170083174e-182)
		tmp = y - ((z / t) * (y - x));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] / 1.0), $MachinePrecision] * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[a, -1.6153062845442575e-142], t$95$1, If[Less[a, 3.774403170083174e-182], N[(y - N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024235 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:linMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< a -646122513817703/4000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))) (if (< a 1887201585041587/50000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- y (* (/ z t) (- y x))) (+ x (* (/ (- y x) 1) (/ (- z t) (- a t)))))))

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