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

Percentage Accurate: 68.3% → 88.6%

Time: 12.8s

Alternatives: 15

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: 68.3% 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: 88.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{+80}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+248}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -2.5e+80)
     t_1
     (if (<= t 1.9e+248) (fma (/ (- z t) (- a t)) (- y x) 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 <= -2.5e+80) {
		tmp = t_1;
	} else if (t <= 1.9e+248) {
		tmp = fma(((z - t) / (a - t)), (y - x), 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 <= -2.5e+80)
		tmp = t_1;
	elseif (t <= 1.9e+248)
		tmp = fma(Float64(Float64(z - t) / Float64(a - t)), Float64(y - x), 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, -2.5e+80], t$95$1, If[LessEqual[t, 1.9e+248], N[(N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.4999999999999998e80 or 1.9e248 < t

   1. Initial program 20.1%

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

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

   5. Applied rewrites 93.9%

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

   if -2.4999999999999998e80 < t < 1.9e248

   1. Initial program 79.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 90.6

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

   4. Applied rewrites 90.6%

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

Alternative 2: 81.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.36 \cdot 10^{+73}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.4 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a - t}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.36e+73)
     t_1
     (if (<= t 3.4e+91) (fma (/ z (- a t)) (- y x) 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 <= -1.36e+73) {
		tmp = t_1;
	} else if (t <= 3.4e+91) {
		tmp = fma((z / (a - t)), (y - x), 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 <= -1.36e+73)
		tmp = t_1;
	elseif (t <= 3.4e+91)
		tmp = fma(Float64(z / Float64(a - t)), Float64(y - x), 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, -1.36e+73], t$95$1, If[LessEqual[t, 3.4e+91], N[(N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.3599999999999999e73 or 3.4000000000000001e91 < t

   1. Initial program 30.7%

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

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

   5. Applied rewrites 84.0%

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

   if -1.3599999999999999e73 < t < 3.4000000000000001e91

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.3

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

   4. Applied rewrites 93.3%

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

   5. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 83.5

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

   7. Applied rewrites 83.5%

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

Alternative 3: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x - y, \frac{z - a}{t}, y\right)\\ \mathbf{if}\;t \leq -1.4 \cdot 10^{-14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.1 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- x y) (/ (- z a) t) y)))
   (if (<= t -1.4e-14) t_1 (if (<= t 3.1e+91) (fma (/ z a) (- y x) 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 <= -1.4e-14) {
		tmp = t_1;
	} else if (t <= 3.1e+91) {
		tmp = fma((z / a), (y - x), 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 <= -1.4e-14)
		tmp = t_1;
	elseif (t <= 3.1e+91)
		tmp = fma(Float64(z / a), Float64(y - x), 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, -1.4e-14], t$95$1, If[LessEqual[t, 3.1e+91], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1.4e-14 or 3.09999999999999998e91 < t

   1. Initial program 37.3%

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

   3. Taylor expanded in t around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. 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. lower-fma.f64 N/A

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

   5. Applied rewrites 79.3%

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

   if -1.4e-14 < t < 3.09999999999999998e91

   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. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 94.3

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

   4. Applied rewrites 94.3%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 74.9

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

   7. Applied rewrites 74.9%

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

Alternative 4: 68.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{if}\;a \leq -7.2 \cdot 10^{-50}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-87}:\\ \;\;\;\;y + \frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z a) (- y x) x)))
   (if (<= a -7.2e-50) t_1 (if (<= a 9.5e-87) (+ y (/ (* z (- x y)) t)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / a), (y - x), x);
	double tmp;
	if (a <= -7.2e-50) {
		tmp = t_1;
	} else if (a <= 9.5e-87) {
		tmp = y + ((z * (x - y)) / t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / a), Float64(y - x), x)
	tmp = 0.0
	if (a <= -7.2e-50)
		tmp = t_1;
	elseif (a <= 9.5e-87)
		tmp = Float64(y + Float64(Float64(z * Float64(x - y)) / t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -7.2e-50], t$95$1, If[LessEqual[a, 9.5e-87], N[(y + N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -7.19999999999999958e-50 or 9.5e-87 < a

   1. Initial program 68.6%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 87.9

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

   4. Applied rewrites 87.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 70.3

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

   7. Applied rewrites 70.3%

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

   if -7.19999999999999958e-50 < a < 9.5e-87

   1. Initial program 63.6%

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

      1. lift-*.f64 N/A

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

      2. lift--.f64 N/A

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

      3. sub-neg N/A

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

      4. +-commutative N/A

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

      5. distribute-lft-in N/A

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

      6. lower-fma.f64 N/A

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

      7. lower-neg.f64 N/A

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

      8. lower-*.f64 63.6

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

   4. Applied rewrites 63.6%

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

   5. Taylor expanded in a around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 78.9

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

   7. Applied rewrites 78.9%

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

4. Final simplification 73.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.2 \cdot 10^{-50}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{elif}\;a \leq 9.5 \cdot 10^{-87}:\\ \;\;\;\;y + \frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 63.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, -y, y\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{a}, y - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.4e+19)
   (fma (/ z t) (- y) y)
   (if (<= t 4.2e+91) (fma (/ z a) (- y x) x) (fma a (/ (- y x) t) y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -4.4e+19) {
		tmp = fma((z / t), -y, y);
	} else if (t <= 4.2e+91) {
		tmp = fma((z / a), (y - x), x);
	} else {
		tmp = fma(a, ((y - x) / t), y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -4.4e+19)
		tmp = fma(Float64(z / t), Float64(-y), y);
	elseif (t <= 4.2e+91)
		tmp = fma(Float64(z / a), Float64(y - x), 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, -4.4e+19], N[(N[(z / t), $MachinePrecision] * (-y) + y), $MachinePrecision], If[LessEqual[t, 4.2e+91], N[(N[(z / a), $MachinePrecision] * N[(y - x), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.4e19

   1. Initial program 35.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 44.0

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 12.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.0%

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

   if -4.4e19 < t < 4.20000000000000015e91

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 73.8

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

   7. Applied rewrites 73.8%

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

   if 4.20000000000000015e91 < t

   1. Initial program 36.4%

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

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 47.0

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

   5. Applied rewrites 47.0%

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

   6. Taylor expanded in t around -inf

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

   8. Applied rewrites 56.9%

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

Alternative 6: 62.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{+19}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, -y, y\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -4.4e+19)
   (fma (/ z t) (- y) y)
   (if (<= t 4.2e+91) (fma z (/ (- y x) 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 <= -4.4e+19) {
		tmp = fma((z / t), -y, y);
	} else if (t <= 4.2e+91) {
		tmp = fma(z, ((y - x) / 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 <= -4.4e+19)
		tmp = fma(Float64(z / t), Float64(-y), y);
	elseif (t <= 4.2e+91)
		tmp = fma(z, Float64(Float64(y - x) / 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, -4.4e+19], N[(N[(z / t), $MachinePrecision] * (-y) + y), $MachinePrecision], If[LessEqual[t, 4.2e+91], N[(z * N[(N[(y - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -4.4e19

   1. Initial program 35.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 44.0

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 12.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.0%

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

   if -4.4e19 < t < 4.20000000000000015e91

   1. Initial program 85.5%

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

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 71.1

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

   5. Applied rewrites 71.1%

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

   if 4.20000000000000015e91 < t

   1. Initial program 36.4%

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

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 47.0

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

   5. Applied rewrites 47.0%

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

   6. Taylor expanded in t around -inf

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

   8. Applied rewrites 56.9%

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

Alternative 7: 56.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -9.5 \cdot 10^{+17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, -y, y\right)\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{+91}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(a, \frac{y - x}{t}, y\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -9.5e+17)
   (fma (/ z t) (- y) y)
   (if (<= t 4.2e+91) (fma z (/ 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 <= -9.5e+17) {
		tmp = fma((z / t), -y, y);
	} else if (t <= 4.2e+91) {
		tmp = fma(z, (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 <= -9.5e+17)
		tmp = fma(Float64(z / t), Float64(-y), y);
	elseif (t <= 4.2e+91)
		tmp = fma(z, Float64(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, -9.5e+17], N[(N[(z / t), $MachinePrecision] * (-y) + y), $MachinePrecision], If[LessEqual[t, 4.2e+91], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(a * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] + y), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.5e17

   1. Initial program 35.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 44.0

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

   5. Applied rewrites 44.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 12.6%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 65.0%

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

   if -9.5e17 < t < 4.20000000000000015e91

   1. Initial program 85.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 93.8

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

   4. Applied rewrites 93.8%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 71.1

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

   7. Applied rewrites 71.1%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 55.4%

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

   if 4.20000000000000015e91 < t

   1. Initial program 36.4%

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

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 47.0

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

   5. Applied rewrites 47.0%

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

   6. Taylor expanded in t around -inf

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

   8. Applied rewrites 56.9%

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

Alternative 8: 56.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -9.2 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.15 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, -y, y\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y a) x)))
   (if (<= a -9.2e+23) t_1 (if (<= a 2.15e-70) (fma (/ z t) (- y) y) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / a), x);
	double tmp;
	if (a <= -9.2e+23) {
		tmp = t_1;
	} else if (a <= 2.15e-70) {
		tmp = fma((z / t), -y, y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / a), x)
	tmp = 0.0
	if (a <= -9.2e+23)
		tmp = t_1;
	elseif (a <= 2.15e-70)
		tmp = fma(Float64(z / t), Float64(-y), y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -9.2e+23], t$95$1, If[LessEqual[a, 2.15e-70], N[(N[(z / t), $MachinePrecision] * (-y) + y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.2000000000000002e23 or 2.15e-70 < a

   1. Initial program 69.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 88.1

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

   4. Applied rewrites 88.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 70.5

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

   7. Applied rewrites 70.5%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 60.8%

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

   if -9.2000000000000002e23 < a < 2.15e-70

   1. Initial program 63.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 53.0

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

   5. Applied rewrites 53.0%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 30.3%

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

   9. Taylor expanded in x around 0

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

   11. Applied rewrites 53.4%

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

Alternative 9: 48.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{if}\;a \leq -1.05 \cdot 10^{-63}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.85 \cdot 10^{-171}:\\ \;\;\;\;\frac{z \cdot \left(x - y\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y a) x)))
   (if (<= a -1.05e-63) t_1 (if (<= a 1.85e-171) (/ (* z (- x y)) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / a), x);
	double tmp;
	if (a <= -1.05e-63) {
		tmp = t_1;
	} else if (a <= 1.85e-171) {
		tmp = (z * (x - y)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / a), x)
	tmp = 0.0
	if (a <= -1.05e-63)
		tmp = t_1;
	elseif (a <= 1.85e-171)
		tmp = Float64(Float64(z * Float64(x - y)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.05e-63], t$95$1, If[LessEqual[a, 1.85e-171], N[(N[(z * N[(x - y), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.05e-63 or 1.85000000000000006e-171 < a

   1. Initial program 68.8%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 87.3

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

   4. Applied rewrites 87.3%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 65.2

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

   7. Applied rewrites 65.2%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 54.8%

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

   if -1.05e-63 < a < 1.85000000000000006e-171

   1. Initial program 62.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 56.2

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

   5. Applied rewrites 56.2%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 49.8%

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

Alternative 10: 52.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.75 \cdot 10^{+45}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.75e+45) y (if (<= t 2.8e+134) (fma z (/ y a) x) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.75e+45) {
		tmp = y;
	} else if (t <= 2.8e+134) {
		tmp = fma(z, (y / a), x);
	} else {
		tmp = y;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.75e+45)
		tmp = y;
	elseif (t <= 2.8e+134)
		tmp = fma(z, Float64(y / a), x);
	else
		tmp = y;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.75e+45], y, If[LessEqual[t, 2.8e+134], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -1.75000000000000011e45 or 2.7999999999999999e134 < 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 a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 42.5

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

   5. Applied rewrites 42.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 14.3%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 51.0%

       \[\leadsto y \]

   if -1.75000000000000011e45 < t < 2.7999999999999999e134

   1. Initial program 83.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 92.7

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

   4. Applied rewrites 92.7%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 67.9

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

   7. Applied rewrites 67.9%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 53.4%

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

Alternative 11: 36.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -1.3 \cdot 10^{-55}:\\ \;\;\;\;y\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{-20}:\\ \;\;\;\;y \cdot \frac{z}{a}\\ \mathbf{else}:\\ \;\;\;\;y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= t -1.3e-55) y (if (<= t 1.05e-20) (* y (/ z a)) y)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (t <= -1.3e-55) {
		tmp = y;
	} else if (t <= 1.05e-20) {
		tmp = 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 <= (-1.3d-55)) then
        tmp = y
    else if (t <= 1.05d-20) then
        tmp = 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 <= -1.3e-55) {
		tmp = y;
	} else if (t <= 1.05e-20) {
		tmp = y * (z / a);
	} else {
		tmp = y;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if t <= -1.3e-55:
		tmp = y
	elif t <= 1.05e-20:
		tmp = y * (z / a)
	else:
		tmp = y
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (t <= -1.3e-55)
		tmp = y;
	elseif (t <= 1.05e-20)
		tmp = Float64(y * Float64(z / a));
	else
		tmp = y;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (t <= -1.3e-55)
		tmp = y;
	elseif (t <= 1.05e-20)
		tmp = y * (z / a);
	else
		tmp = y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[t, -1.3e-55], y, If[LessEqual[t, 1.05e-20], N[(y * N[(z / a), $MachinePrecision]), $MachinePrecision], y]]

Derivation

1. Split input into 2 regimes

2. if t < -1.2999999999999999e-55 or 1.0499999999999999e-20 < t

   1. Initial program 47.5%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 39.9

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

   5. Applied rewrites 39.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 17.2%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 38.8%

       \[\leadsto y \]

   if -1.2999999999999999e-55 < t < 1.0499999999999999e-20

   1. Initial program 88.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. *-commutative N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 95.1

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

   4. Applied rewrites 95.1%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 75.4

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

   7. Applied rewrites 75.4%

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

   8. Taylor expanded in y around inf

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

   10. Applied rewrites 29.8%

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

Alternative 12: 29.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+174}:\\ \;\;\;\;z \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+139}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= x -1.65e+174) (* z (/ x t)) (if (<= x 6.2e+139) y (* x (/ z t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.65e+174) {
		tmp = z * (x / t);
	} else if (x <= 6.2e+139) {
		tmp = y;
	} else {
		tmp = x * (z / t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (x <= (-1.65d+174)) then
        tmp = z * (x / t)
    else if (x <= 6.2d+139) then
        tmp = y
    else
        tmp = x * (z / t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (x <= -1.65e+174) {
		tmp = z * (x / t);
	} else if (x <= 6.2e+139) {
		tmp = y;
	} else {
		tmp = x * (z / t);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if x <= -1.65e+174:
		tmp = z * (x / t)
	elif x <= 6.2e+139:
		tmp = y
	else:
		tmp = x * (z / t)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (x <= -1.65e+174)
		tmp = Float64(z * Float64(x / t));
	elseif (x <= 6.2e+139)
		tmp = y;
	else
		tmp = Float64(x * Float64(z / t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (x <= -1.65e+174)
		tmp = z * (x / t);
	elseif (x <= 6.2e+139)
		tmp = y;
	else
		tmp = x * (z / t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[x, -1.65e+174], N[(z * N[(x / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 6.2e+139], y, N[(x * N[(z / t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.65e174

   1. Initial program 62.6%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 33.2

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

   5. Applied rewrites 33.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 36.8%

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

   10. Applied rewrites 45.8%

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

   if -1.65e174 < x < 6.2e139

   1. Initial program 68.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 11.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 29.5%

       \[\leadsto y \]

   if 6.2e139 < x

   1. Initial program 60.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 29.4

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

   5. Applied rewrites 29.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 36.4%

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

   10. Applied rewrites 44.6%

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

4. Final simplification 33.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1.65 \cdot 10^{+174}:\\ \;\;\;\;z \cdot \frac{x}{t}\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+139}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{z}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 29.7% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{x}{t}\\ \mathbf{if}\;x \leq -1.65 \cdot 10^{+174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 6.2 \cdot 10^{+139}:\\ \;\;\;\;y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* z (/ x t))))
   (if (<= x -1.65e+174) t_1 (if (<= x 6.2e+139) y t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (x / t);
	double tmp;
	if (x <= -1.65e+174) {
		tmp = t_1;
	} else if (x <= 6.2e+139) {
		tmp = y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = z * (x / t)
    if (x <= (-1.65d+174)) then
        tmp = t_1
    else if (x <= 6.2d+139) then
        tmp = y
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = z * (x / t);
	double tmp;
	if (x <= -1.65e+174) {
		tmp = t_1;
	} else if (x <= 6.2e+139) {
		tmp = y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = z * (x / t)
	tmp = 0
	if x <= -1.65e+174:
		tmp = t_1
	elif x <= 6.2e+139:
		tmp = y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(z * Float64(x / t))
	tmp = 0.0
	if (x <= -1.65e+174)
		tmp = t_1;
	elseif (x <= 6.2e+139)
		tmp = y;
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = z * (x / t);
	tmp = 0.0;
	if (x <= -1.65e+174)
		tmp = t_1;
	elseif (x <= 6.2e+139)
		tmp = y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(x / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[x, -1.65e+174], t$95$1, If[LessEqual[x, 6.2e+139], y, t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.65e174 or 6.2e139 < x

   1. Initial program 61.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 31.2

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

   5. Applied rewrites 31.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 36.6%

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

   10. Applied rewrites 39.3%

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

   if -1.65e174 < x < 6.2e139

   1. Initial program 68.4%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. lower-fma.f64 N/A

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

      7. lower--.f64 N/A

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

      8. distribute-neg-frac N/A

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

      9. mul-1-neg N/A

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

      10. lower-/.f64 N/A

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

      11. mul-1-neg N/A

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

      12. sub-neg N/A

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

      13. +-commutative N/A

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

      14. distribute-neg-in N/A

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

      15. unsub-neg N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 34.1

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

   5. Applied rewrites 34.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 11.7%

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

   9. Taylor expanded in z around 0

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

   11. Applied rewrites 29.5%

       \[\leadsto y \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 14: 25.9% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 66.6%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. mul-1-neg N/A

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

   3. *-commutative N/A

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

   4. associate-/l* N/A

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

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

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

   6. lower-fma.f64 N/A

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

   7. lower--.f64 N/A

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

   8. distribute-neg-frac N/A

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

   9. mul-1-neg N/A

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

   10. lower-/.f64 N/A

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

   11. mul-1-neg N/A

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

   12. sub-neg N/A

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

   13. +-commutative N/A

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

   14. distribute-neg-in N/A

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

   15. unsub-neg N/A

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

   16. remove-double-neg N/A

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

   17. lower--.f64 33.4

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

5. Applied rewrites 33.4%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 18.0%

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

9. Taylor expanded in z around 0

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

11. Applied rewrites 23.9%

    \[\leadsto y \]
12. Add Preprocessing

Alternative 15: 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 66.6%

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

3. Taylor expanded in x around inf

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

   1. +-commutative N/A

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

   2. distribute-rgt-in N/A

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

   3. mul-1-neg N/A

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

   4. distribute-lft-neg-out N/A

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

   5. *-commutative N/A

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

   6. associate-/l* N/A

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

   7. *-commutative N/A

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

   8. associate-/l* N/A

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

   9. distribute-rgt-neg-in N/A

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

   10. *-lft-identity N/A

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

   11. lower-fma.f64 N/A

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

   12. lower--.f64 N/A

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

   13. distribute-neg-frac2 N/A

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

   14. lower-/.f64 N/A

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

   15. sub-neg N/A

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

   16. +-commutative N/A

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

   17. distribute-neg-in N/A

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

   18. remove-double-neg N/A

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

   19. lower-+.f64 N/A

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

   20. lower-neg.f64 45.4

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

5. Applied rewrites 45.4%

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

6. Taylor expanded in t around inf

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

8. Applied rewrites 2.8%

   \[\leadsto 0 \]
9. 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 2024220 
(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))))