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

Percentage Accurate: 68.2% → 91.2%

Time: 13.6s

Alternatives: 16

Speedup: 0.8×

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: 68.2% 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: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-299}:\\ \;\;\;\;x + \frac{y - x}{\frac{a - t}{z - t}}\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_1 -5e-299)
     (+ x (/ (- y x) (/ (- a t) (- z t))))
     (if (<= t_1 0.0)
       (fma (- x y) (/ (- z a) t) y)
       (fma (/ (- z t) (- a t)) (- y x) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_1 <= -5e-299) {
		tmp = x + ((y - x) / ((a - t) / (z - t)));
	} else if (t_1 <= 0.0) {
		tmp = fma((x - y), ((z - a) / t), y);
	} else {
		tmp = fma(((z - t) / (a - t)), (y - x), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= -5e-299)
		tmp = Float64(x + Float64(Float64(y - x) / Float64(Float64(a - t) / Float64(z - t))));
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(x - y), Float64(Float64(z - a) / t), y);
	else
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-299], N[(x + N[(N[(y - x), $MachinePrecision] / N[(N[(a - t), $MachinePrecision] / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < -4.99999999999999956e-299

   1. Initial program 73.2%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 92.4

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

   4. Applied egg-rr 92.4%

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

   if -4.99999999999999956e-299 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   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. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

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

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

   1. Initial program 73.2%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 89.8

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

   4. Applied egg-rr 89.8%

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

Alternative 2: 91.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ t_2 := x + \frac{\left(y - x\right) \cdot \left(z - t\right)}{a - t}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-299}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z t) (- a t)) (- y x) x))
        (t_2 (+ x (/ (* (- y x) (- z t)) (- a t)))))
   (if (<= t_2 -5e-299)
     t_1
     (if (<= t_2 0.0) (fma (- x y) (/ (- z a) t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - t) / (a - t)), (y - x), x);
	double t_2 = x + (((y - x) * (z - t)) / (a - t));
	double tmp;
	if (t_2 <= -5e-299) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma((x - y), ((z - a) / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), x)
	t_2 = Float64(x + Float64(Float64(Float64(y - x) * Float64(z - t)) / Float64(a - t)))
	tmp = 0.0
	if (t_2 <= -5e-299)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(x - y), Float64(Float64(z - a) / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(N[(y - x), $MachinePrecision] * N[(z - t), $MachinePrecision]), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-299], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $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))) < -4.99999999999999956e-299 or 0.0 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t)))

   1. Initial program 73.2%

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. /-lowering-/.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 90.9

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

   4. Applied egg-rr 90.9%

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

   if -4.99999999999999956e-299 < (+.f64 x (/.f64 (*.f64 (-.f64 y x) (-.f64 z t)) (-.f64 a t))) < 0.0

   1. Initial program 4.8%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   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. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 99.9%

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

Alternative 3: 63.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t \leq -8 \cdot 10^{+40}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -2.45 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{z}{a - t}, x\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t)))))
   (if (<= t -8e+40)
     t_1
     (if (<= t -2.45e-28)
       (fma (- x) (/ z (- a t)) x)
       (if (<= t 5.4e-48) (+ x (/ (* (- y x) z) a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t <= -8e+40) {
		tmp = t_1;
	} else if (t <= -2.45e-28) {
		tmp = fma(-x, (z / (a - t)), x);
	} else if (t <= 5.4e-48) {
		tmp = x + (((y - x) * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	tmp = 0.0
	if (t <= -8e+40)
		tmp = t_1;
	elseif (t <= -2.45e-28)
		tmp = fma(Float64(-x), Float64(z / Float64(a - t)), x);
	elseif (t <= 5.4e-48)
		tmp = Float64(x + Float64(Float64(Float64(y - x) * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -8e+40], t$95$1, If[LessEqual[t, -2.45e-28], N[((-x) * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 5.4e-48], N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -8.00000000000000024e40 or 5.40000000000000023e-48 < t

   1. Initial program 37.8%

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

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 38.7

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

   5. Simplified 38.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 67.1

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

   7. Applied egg-rr 67.1%

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

   if -8.00000000000000024e40 < t < -2.45000000000000015e-28

   1. Initial program 83.1%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. --lowering--.f64 N/A

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

      7. neg-lowering-neg.f64 N/A

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

      8. *-lowering-*.f64 N/A

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

      9. --lowering--.f64 83.1

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

   4. Applied egg-rr 83.1%

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

   5. Taylor expanded in z around inf

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

   7. Simplified 67.1%

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

   8. Taylor expanded in z around inf

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

   10. Simplified 69.2%

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

   11. Taylor expanded in x around inf

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

       1. +-commutative N/A

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

       2. distribute-lft-in N/A

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

       3. associate-*r* N/A

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

       4. *-commutative N/A

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

       5. *-rgt-identity N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. mul-1-neg N/A

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

       8. neg-lowering-neg.f64 N/A

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

       9. /-lowering-/.f64 N/A

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

       10. --lowering--.f64 63.9

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

   13. Simplified 63.9%

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

   if -2.45000000000000015e-28 < t < 5.40000000000000023e-48

   1. Initial program 90.0%

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

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 72.5

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

   5. Simplified 72.5%

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

4. Final simplification 69.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+40}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq -2.45 \cdot 10^{-28}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{z}{a - t}, x\right)\\ \mathbf{elif}\;t \leq 5.4 \cdot 10^{-48}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 60.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-z}{t}, y\right)\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x - y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -9e+45)
   (fma y (/ (- z) t) y)
   (if (<= t 5.2e-86)
     (fma z (/ (- y x) a) x)
     (if (<= t 6.1e+140) (fma t (/ (- x y) a) x) (fma a (/ (- y x) t) y)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -9e+45) {
		tmp = fma(y, (-z / t), y);
	} else if (t <= 5.2e-86) {
		tmp = fma(z, ((y - x) / a), x);
	} else if (t <= 6.1e+140) {
		tmp = fma(t, ((x - y) / a), x);
	} else {
		tmp = fma(a, ((y - x) / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -9e+45)
		tmp = fma(y, Float64(Float64(-z) / t), y);
	elseif (t <= 5.2e-86)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	elseif (t <= 6.1e+140)
		tmp = fma(t, Float64(Float64(x - y) / a), x);
	else
		tmp = fma(a, Float64(Float64(y - x) / t), y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -9e+45], N[(y * N[((-z) / t), $MachinePrecision] + y), $MachinePrecision], If[LessEqual[t, 5.2e-86], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 6.1e+140], N[(t * N[(N[(x - y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if t < -8.9999999999999997e45

   1. Initial program 36.8%

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

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 36.5

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

   5. Simplified 36.5%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. div-sub N/A

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

      6. *-inverses N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 60.5

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

   8. Simplified 60.5%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. *-rgt-identity N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. mul-1-neg N/A

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

       8. distribute-neg-frac2 N/A

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

       9. mul-1-neg N/A

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

       10. /-lowering-/.f64 N/A

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

       11. mul-1-neg N/A

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

       12. neg-lowering-neg.f64 60.5

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

   11. Simplified 60.5%

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

   if -8.9999999999999997e45 < t < 5.2000000000000002e-86

   1. Initial program 88.5%

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

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 69.1

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

   5. Simplified 69.1%

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

   if 5.2000000000000002e-86 < t < 6.0999999999999996e140

   1. Initial program 73.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. --lowering--.f64 55.8

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

   5. Simplified 55.8%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 59.5

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

   8. Simplified 59.5%

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

   if 6.0999999999999996e140 < t

   1. Initial program 23.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. --lowering--.f64 42.8

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

   5. Simplified 42.8%

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

   6. Taylor expanded in a around 0

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. distribute-rgt1-in N/A

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

      14. metadata-eval N/A

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

      15. mul0-lft N/A

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

      18. cancel-sign-sub-inv N/A

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

      19. neg-sub0 N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

      22. div-sub N/A

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

      23. associate-/l* N/A

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

   8. Simplified 66.8%

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

4. Final simplification 65.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -9 \cdot 10^{+45}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-z}{t}, y\right)\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-86}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x - y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 52.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{if}\;t \leq -6.5 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-86}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x - y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- y x) t) y)))
   (if (<= t -6.5e+106)
     t_1
     (if (<= t 9e-86)
       (+ x (/ (* y z) a))
       (if (<= t 6.1e+140) (fma t (/ (- x y) a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((y - x) / t), y);
	double tmp;
	if (t <= -6.5e+106) {
		tmp = t_1;
	} else if (t <= 9e-86) {
		tmp = x + ((y * z) / a);
	} else if (t <= 6.1e+140) {
		tmp = fma(t, ((x - y) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(y - x) / t), y)
	tmp = 0.0
	if (t <= -6.5e+106)
		tmp = t_1;
	elseif (t <= 9e-86)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	elseif (t <= 6.1e+140)
		tmp = fma(t, Float64(Float64(x - y) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -6.5e+106], t$95$1, If[LessEqual[t, 9e-86], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 6.1e+140], N[(t * N[(N[(x - y), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -6.5000000000000003e106 or 6.0999999999999996e140 < t

   1. Initial program 28.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. --lowering--.f64 52.3

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

   5. Simplified 52.3%

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

   6. Taylor expanded in a around 0

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. distribute-rgt1-in N/A

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

      14. metadata-eval N/A

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

      15. mul0-lft N/A

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

      18. cancel-sign-sub-inv N/A

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

      19. neg-sub0 N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

      22. div-sub N/A

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

      23. associate-/l* N/A

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

   8. Simplified 67.3%

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

   if -6.5000000000000003e106 < t < 8.9999999999999995e-86

   1. Initial program 85.7%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 91.3

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

   4. Applied egg-rr 91.3%

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

   5. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 69.0

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

   7. Simplified 69.0%

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

   8. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 57.4

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

   10. Simplified 57.4%

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

   if 8.9999999999999995e-86 < t < 6.0999999999999996e140

   1. Initial program 73.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. --lowering--.f64 55.8

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

   5. Simplified 55.8%

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

   6. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. div-sub N/A

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

      3. accelerator-lowering-fma.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 59.5

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

   8. Simplified 59.5%

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

4. Final simplification 60.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.5 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-86}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{x - y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 86.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.25 \cdot 10^{+63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+141}:\\ \;\;\;\;\mathsf{fma}\left(z - t, \frac{y - x}{a - t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.25e+63)
     t_1
     (if (<= t 1.6e+141) (fma (- z t) (/ (- y x) (- a t)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x - y), ((z - a) / t), y);
	double tmp;
	if (t <= -1.25e+63) {
		tmp = t_1;
	} else if (t <= 1.6e+141) {
		tmp = fma((z - t), ((y - x) / (a - t)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x - y), Float64(Float64(z - a) / t), y)
	tmp = 0.0
	if (t <= -1.25e+63)
		tmp = t_1;
	elseif (t <= 1.6e+141)
		tmp = fma(Float64(z - t), Float64(Float64(y - x) / Float64(a - t)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.25e+63], t$95$1, If[LessEqual[t, 1.6e+141], N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.25000000000000003e63 or 1.60000000000000009e141 < t

   1. Initial program 29.2%

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 84.6%

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

   if -1.25000000000000003e63 < t < 1.60000000000000009e141

   1. Initial program 86.2%

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 88.6

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

   4. Applied egg-rr 88.6%

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

Alternative 7: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z - t, \frac{y - x}{a}, x\right)\\ \mathbf{if}\;a \leq -1 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.15 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- z t) (/ (- y x) a) x)))
   (if (<= a -1e-72)
     t_1
     (if (<= a 1.15e+21) (fma (- x y) (/ (- z a) t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z - t), ((y - x) / a), x);
	double tmp;
	if (a <= -1e-72) {
		tmp = t_1;
	} else if (a <= 1.15e+21) {
		tmp = fma((x - y), ((z - a) / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z - t), Float64(Float64(y - x) / a), x)
	tmp = 0.0
	if (a <= -1e-72)
		tmp = t_1;
	elseif (a <= 1.15e+21)
		tmp = fma(Float64(x - y), Float64(Float64(z - a) / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1e-72], t$95$1, If[LessEqual[a, 1.15e+21], N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.9999999999999997e-73 or 1.15e21 < a

   1. Initial program 73.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 75.1

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

   5. Simplified 75.1%

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

   if -9.9999999999999997e-73 < a < 1.15e21

   1. Initial program 60.7%

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 78.6%

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

Alternative 8: 73.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z - t, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{+61}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- z t) (/ y a) x)))
   (if (<= a -2.9e+61)
     t_1
     (if (<= a 3.8e+19) (fma (- x y) (/ (- z a) t) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z - t), (y / a), x);
	double tmp;
	if (a <= -2.9e+61) {
		tmp = t_1;
	} else if (a <= 3.8e+19) {
		tmp = fma((x - y), ((z - a) / t), y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z - t), Float64(y / a), x)
	tmp = 0.0
	if (a <= -2.9e+61)
		tmp = t_1;
	elseif (a <= 3.8e+19)
		tmp = fma(Float64(x - y), Float64(Float64(z - a) / t), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.9e+61], t$95$1, If[LessEqual[a, 3.8e+19], N[(N[(x - y), $MachinePrecision] * N[(N[(z - a), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.9000000000000001e61 or 3.8e19 < a

   1. Initial program 73.6%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. accelerator-lowering-fma.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 80.1

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

   5. Simplified 80.1%

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

   6. Taylor expanded in y around inf

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

   8. Simplified 74.9%

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

   if -2.9000000000000001e61 < a < 3.8e19

   1. Initial program 62.9%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\left(y + -1 \cdot \frac{z \cdot \left(y - x\right)}{t}\right) - -1 \cdot \frac{a \cdot \left(y - x\right)}{t}} \]
   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. distribute-rgt-out-- N/A

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. accelerator-lowering-fma.f64 N/A

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

   5. Simplified 73.2%

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

Alternative 9: 63.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t \leq -3.7 \cdot 10^{+43}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-47}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t)))))
   (if (<= t -3.7e+43)
     t_1
     (if (<= t 2.95e-47) (+ x (/ (* (- y x) z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t <= -3.7e+43) {
		tmp = t_1;
	} else if (t <= 2.95e-47) {
		tmp = x + (((y - x) * z) / a);
	} 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 * ((z - t) / (a - t))
    if (t <= (-3.7d+43)) then
        tmp = t_1
    else if (t <= 2.95d-47) then
        tmp = x + (((y - x) * z) / a)
    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 * ((z - t) / (a - t));
	double tmp;
	if (t <= -3.7e+43) {
		tmp = t_1;
	} else if (t <= 2.95e-47) {
		tmp = x + (((y - x) * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * ((z - t) / (a - t))
	tmp = 0
	if t <= -3.7e+43:
		tmp = t_1
	elif t <= 2.95e-47:
		tmp = x + (((y - x) * z) / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	tmp = 0.0
	if (t <= -3.7e+43)
		tmp = t_1;
	elseif (t <= 2.95e-47)
		tmp = Float64(x + Float64(Float64(Float64(y - x) * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * ((z - t) / (a - t));
	tmp = 0.0;
	if (t <= -3.7e+43)
		tmp = t_1;
	elseif (t <= 2.95e-47)
		tmp = x + (((y - x) * z) / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3.7e+43], t$95$1, If[LessEqual[t, 2.95e-47], N[(x + N[(N[(N[(y - x), $MachinePrecision] * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.7000000000000001e43 or 2.94999999999999986e-47 < t

   1. Initial program 37.8%

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

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 38.7

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

   5. Simplified 38.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 67.1

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

   7. Applied egg-rr 67.1%

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

   if -3.7000000000000001e43 < t < 2.94999999999999986e-47

   1. Initial program 89.2%

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

   3. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 68.2

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

   5. Simplified 68.2%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.7 \cdot 10^{+43}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 2.95 \cdot 10^{-47}:\\ \;\;\;\;x + \frac{\left(y - x\right) \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 65.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t \leq -4.3 \cdot 10^{+42}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t)))))
   (if (<= t -4.3e+42) t_1 (if (<= t 7e-50) (fma z (/ (- y x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t <= -4.3e+42) {
		tmp = t_1;
	} else if (t <= 7e-50) {
		tmp = fma(z, ((y - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	tmp = 0.0
	if (t <= -4.3e+42)
		tmp = t_1;
	elseif (t <= 7e-50)
		tmp = fma(z, Float64(Float64(y - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -4.3e+42], t$95$1, If[LessEqual[t, 7e-50], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.2999999999999998e42 or 6.99999999999999993e-50 < t

   1. Initial program 37.8%

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

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 38.7

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

   5. Simplified 38.7%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. --lowering--.f64 67.1

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

   7. Applied egg-rr 67.1%

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

   if -4.2999999999999998e42 < t < 6.99999999999999993e-50

   1. Initial program 89.2%

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

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 67.7

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

   5. Simplified 67.7%

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

4. Final simplification 67.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.3 \cdot 10^{+42}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 53.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+140}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma a (/ (- y x) t) y)))
   (if (<= t -1.4e+106) t_1 (if (<= t 6.1e+140) (+ x (/ (* y z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(a, ((y - x) / t), y);
	double tmp;
	if (t <= -1.4e+106) {
		tmp = t_1;
	} else if (t <= 6.1e+140) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(a, Float64(Float64(y - x) / t), y)
	tmp = 0.0
	if (t <= -1.4e+106)
		tmp = t_1;
	elseif (t <= 6.1e+140)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.4e+106], t$95$1, If[LessEqual[t, 6.1e+140], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.39999999999999996e106 or 6.0999999999999996e140 < t

   1. Initial program 28.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

      14. --lowering--.f64 N/A

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

      15. /-lowering-/.f64 N/A

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

      16. --lowering--.f64 52.3

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

   5. Simplified 52.3%

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

   6. Taylor expanded in a around 0

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

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

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

      2. sub-neg N/A

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

      3. mul-1-neg N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. associate-+l+ N/A

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

      9. sub-neg N/A

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

      10. mul-1-neg N/A

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

      11. +-commutative N/A

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

      12. associate-+l+ N/A

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

      13. distribute-rgt1-in N/A

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

      14. metadata-eval N/A

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

      15. mul0-lft N/A

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

      16. *-lft-identity N/A

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

      17. metadata-eval N/A

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

      18. cancel-sign-sub-inv N/A

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

      19. neg-sub0 N/A

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

      20. mul-1-neg N/A

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

      21. remove-double-neg N/A

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

      22. div-sub N/A

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

      23. associate-/l* N/A

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

   8. Simplified 67.3%

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

   if -1.39999999999999996e106 < t < 6.0999999999999996e140

   1. Initial program 83.8%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 91.0

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

   4. Applied egg-rr 91.0%

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

   5. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 64.8

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

   7. Simplified 64.8%

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

   8. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 54.0

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

   10. Simplified 54.0%

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

4. Final simplification 58.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.4 \cdot 10^{+106}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+140}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 53.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{-z}{t}, y\right)\\ \mathbf{if}\;t \leq -1.6 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+109}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ (- z) t) y)))
   (if (<= t -1.6e+46) t_1 (if (<= t 5e+109) (+ x (/ (* y z) a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (-z / t), y);
	double tmp;
	if (t <= -1.6e+46) {
		tmp = t_1;
	} else if (t <= 5e+109) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(Float64(-z) / t), y)
	tmp = 0.0
	if (t <= -1.6e+46)
		tmp = t_1;
	elseif (t <= 5e+109)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[((-z) / t), $MachinePrecision] + y), $MachinePrecision]}, If[LessEqual[t, -1.6e+46], t$95$1, If[LessEqual[t, 5e+109], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.5999999999999999e46 or 5.0000000000000001e109 < 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 x around 0

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

      1. /-lowering-/.f64 N/A

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

      2. *-lowering-*.f64 N/A

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

      3. --lowering--.f64 N/A

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

      4. --lowering--.f64 37.1

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

   5. Simplified 37.1%

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

   6. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. neg-lowering-neg.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-lowering-*.f64 N/A

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

      5. div-sub N/A

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

      6. *-inverses N/A

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

      7. sub-neg N/A

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

      8. metadata-eval N/A

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

      9. +-lowering-+.f64 N/A

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

      10. /-lowering-/.f64 60.7

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

   8. Simplified 60.7%

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

   9. Taylor expanded in y around 0

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

       1. sub-neg N/A

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

       2. mul-1-neg N/A

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

       3. +-commutative N/A

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

       4. distribute-lft-in N/A

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

       5. *-rgt-identity N/A

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

       6. accelerator-lowering-fma.f64 N/A

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

       7. mul-1-neg N/A

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

       8. distribute-neg-frac2 N/A

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

       9. mul-1-neg N/A

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

       10. /-lowering-/.f64 N/A

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

       11. mul-1-neg N/A

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

       12. neg-lowering-neg.f64 60.7

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

   11. Simplified 60.7%

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

   if -1.5999999999999999e46 < t < 5.0000000000000001e109

   1. Initial program 87.9%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 92.4

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

   4. Applied egg-rr 92.4%

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

   5. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 67.7

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

   7. Simplified 67.7%

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

   8. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 56.3

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

   10. Simplified 56.3%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.6 \cdot 10^{+46}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-z}{t}, y\right)\\ \mathbf{elif}\;t \leq 5 \cdot 10^{+109}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{-z}{t}, y\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+61}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+111}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -8e+61) y (if (<= t 2.25e+111) (+ x (/ (* y z) a)) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8e+61) {
		tmp = y;
	} else if (t <= 2.25e+111) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-8d+61)) then
        tmp = y
    else if (t <= 2.25d+111) then
        tmp = x + ((y * z) / a)
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -8e+61) {
		tmp = y;
	} else if (t <= 2.25e+111) {
		tmp = x + ((y * z) / a);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -8e+61:
		tmp = y
	elif t <= 2.25e+111:
		tmp = x + ((y * z) / a)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -8e+61)
		tmp = y;
	elseif (t <= 2.25e+111)
		tmp = Float64(x + Float64(Float64(y * z) / a));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -8e+61)
		tmp = y;
	elseif (t <= 2.25e+111)
		tmp = x + ((y * z) / a);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -8e+61], y, If[LessEqual[t, 2.25e+111], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -7.9999999999999996e61 or 2.25e111 < t

   1. Initial program 29.0%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Simplified 54.9%

      \[\leadsto \color{blue}{y} \]

   if -7.9999999999999996e61 < t < 2.25e111

   1. Initial program 88.1%

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

      1. associate-/l* N/A

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

      2. clear-num N/A

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

      3. un-div-inv N/A

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

      4. /-lowering-/.f64 N/A

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

      5. --lowering--.f64 N/A

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. --lowering--.f64 92.6

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

   4. Applied egg-rr 92.6%

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

   5. Taylor expanded in t around 0

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

      1. /-lowering-/.f64 67.1

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

   7. Simplified 67.1%

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

   8. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 56.0

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

   10. Simplified 56.0%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -8 \cdot 10^{+61}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+111}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 38.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -5.6 \cdot 10^{+45}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 6.1 \cdot 10^{+140}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -5.6e+45) y (if (<= t 6.1e+140) x y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -5.6e+45) {
		tmp = y;
	} else if (t <= 6.1e+140) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (t <= (-5.6d+45)) then
        tmp = y
    else if (t <= 6.1d+140) then
        tmp = x
    else
        tmp = y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -5.6e+45) {
		tmp = y;
	} else if (t <= 6.1e+140) {
		tmp = x;
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -5.6e+45:
		tmp = y
	elif t <= 6.1e+140:
		tmp = x
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -5.6e+45)
		tmp = y;
	elseif (t <= 6.1e+140)
		tmp = x;
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -5.6e+45)
		tmp = y;
	elseif (t <= 6.1e+140)
		tmp = x;
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -5.6e+45], y, If[LessEqual[t, 6.1e+140], x, y]]

Derivation

1. Split input into 2 regimes

2. if t < -5.5999999999999999e45 or 6.0999999999999996e140 < t

   1. Initial program 31.6%

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

   3. Taylor expanded in t around inf

      \[\leadsto \color{blue}{y} \]
   4. Step-by-step derivation

   5. Simplified 56.2%

      \[\leadsto \color{blue}{y} \]

   if -5.5999999999999999e45 < t < 6.0999999999999996e140

   1. Initial program 86.0%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 38.5%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 15: 25.3% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

3. Taylor expanded in a around inf

   \[\leadsto \color{blue}{x} \]
4. Step-by-step derivation

5. Simplified 28.8%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Alternative 16: 2.8% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = 0.0d0
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 67.3%

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

   1. *-commutative N/A

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

   2. sub-neg N/A

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

   3. +-commutative N/A

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

   4. distribute-lft-in N/A

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

   5. accelerator-lowering-fma.f64 N/A

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

   6. --lowering--.f64 N/A

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

   7. neg-lowering-neg.f64 N/A

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

   8. *-lowering-*.f64 N/A

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

   9. --lowering--.f64 66.9

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

4. Applied egg-rr 66.9%

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

5. Taylor expanded in z around inf

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

7. Simplified 56.5%

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

8. Taylor expanded in t around inf

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

   1. distribute-rgt1-in N/A

      \[\leadsto \color{blue}{\left(-1 + 1\right) \cdot x} \]

   2. metadata-eval N/A

      \[\leadsto \color{blue}{0} \cdot x \]

   3. mul0-lft 2.8

      \[\leadsto \color{blue}{0} \]

10. Simplified 2.8%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 87.0% 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 2024198 
(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))))