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

Percentage Accurate: 68.9% → 85.2%

Time: 12.5s

Alternatives: 18

Speedup: 0.9×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 68.9% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 85.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.75 \cdot 10^{-123}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a - z}, y - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.75e-123)
   (fma (- t x) (/ (- y z) (- a z)) x)
   (if (<= a 9.2e-112)
     (fma (- x t) (/ y z) t)
     (fma (/ (- t x) (- a z)) (- y z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.75e-123) {
		tmp = fma((t - x), ((y - z) / (a - z)), x);
	} else if (a <= 9.2e-112) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = fma(((t - x) / (a - z)), (y - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.75e-123)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x);
	elseif (a <= 9.2e-112)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = fma(Float64(Float64(t - x) / Float64(a - z)), Float64(y - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.75e-123], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 9.2e-112], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.7499999999999999e-123

   1. Initial program 81.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 98.8

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

   4. Applied egg-rr 98.8%

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

   if -1.7499999999999999e-123 < a < 9.19999999999999963e-112

   1. Initial program 61.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 93.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 93.7

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

   8. Simplified 93.7%

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

   if 9.19999999999999963e-112 < a

   1. Initial program 72.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 88.1

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

   4. Applied egg-rr 88.1%

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

Alternative 2: 33.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{a}{z}, t\right)\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-100}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t + \left(x - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ y z))))
   (if (<= z -1.1e+162)
     (fma t (/ a z) t)
     (if (<= z -1.02e-151)
       t_1
       (if (<= z 8.4e-100)
         (* t (/ y a))
         (if (<= z 4.9e+143) t_1 (+ t (- x x))))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y / z);
	double tmp;
	if (z <= -1.1e+162) {
		tmp = fma(t, (a / z), t);
	} else if (z <= -1.02e-151) {
		tmp = t_1;
	} else if (z <= 8.4e-100) {
		tmp = t * (y / a);
	} else if (z <= 4.9e+143) {
		tmp = t_1;
	} else {
		tmp = t + (x - x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(y / z))
	tmp = 0.0
	if (z <= -1.1e+162)
		tmp = fma(t, Float64(a / z), t);
	elseif (z <= -1.02e-151)
		tmp = t_1;
	elseif (z <= 8.4e-100)
		tmp = Float64(t * Float64(y / a));
	elseif (z <= 4.9e+143)
		tmp = t_1;
	else
		tmp = Float64(t + Float64(x - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+162], N[(t * N[(a / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, -1.02e-151], t$95$1, If[LessEqual[z, 8.4e-100], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.9e+143], t$95$1, N[(t + N[(x - x), $MachinePrecision]), $MachinePrecision]]]]]]

Derivation

1. Split input into 4 regimes

2. if z < -1.1000000000000001e162

   1. Initial program 46.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 85.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. div-sub N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-neg.f64 70.5

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

   8. Simplified 70.5%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 63.3

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

   11. Simplified 63.3%

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

   if -1.1000000000000001e162 < z < -1.0200000000000001e-151 or 8.40000000000000037e-100 < z < 4.89999999999999986e143

   1. Initial program 77.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 53.4%

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

   6. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 33.1

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

   8. Simplified 33.1%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 31.4

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

   11. Simplified 31.4%

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

   if -1.0200000000000001e-151 < z < 8.40000000000000037e-100

   1. Initial program 91.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 93.7

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

   5. Simplified 93.7%

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

   6. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 38.6

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

   8. Simplified 38.6%

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

   if 4.89999999999999986e143 < z

   1. Initial program 22.4%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 51.0

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

   5. Simplified 51.0%

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

      1. lift--.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 65.6

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

   7. Applied egg-rr 65.6%

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

4. Final simplification 40.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+162}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{a}{z}, t\right)\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-151}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-100}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+143}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t + \left(x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 33.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{z}\\ t_2 := t + \left(x - x\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+162}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-151}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-100}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+143}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* x (/ y z))) (t_2 (+ t (- x x))))
   (if (<= z -1.1e+162)
     t_2
     (if (<= z -1.02e-151)
       t_1
       (if (<= z 8.4e-100) (* t (/ y a)) (if (<= z 4.9e+143) t_1 t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x * (y / z);
	double t_2 = t + (x - x);
	double tmp;
	if (z <= -1.1e+162) {
		tmp = t_2;
	} else if (z <= -1.02e-151) {
		tmp = t_1;
	} else if (z <= 8.4e-100) {
		tmp = t * (y / a);
	} else if (z <= 4.9e+143) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	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) :: t_2
    real(8) :: tmp
    t_1 = x * (y / z)
    t_2 = t + (x - x)
    if (z <= (-1.1d+162)) then
        tmp = t_2
    else if (z <= (-1.02d-151)) then
        tmp = t_1
    else if (z <= 8.4d-100) then
        tmp = t * (y / a)
    else if (z <= 4.9d+143) then
        tmp = t_1
    else
        tmp = t_2
    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 / z);
	double t_2 = t + (x - x);
	double tmp;
	if (z <= -1.1e+162) {
		tmp = t_2;
	} else if (z <= -1.02e-151) {
		tmp = t_1;
	} else if (z <= 8.4e-100) {
		tmp = t * (y / a);
	} else if (z <= 4.9e+143) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x * (y / z)
	t_2 = t + (x - x)
	tmp = 0
	if z <= -1.1e+162:
		tmp = t_2
	elif z <= -1.02e-151:
		tmp = t_1
	elif z <= 8.4e-100:
		tmp = t * (y / a)
	elif z <= 4.9e+143:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x * Float64(y / z))
	t_2 = Float64(t + Float64(x - x))
	tmp = 0.0
	if (z <= -1.1e+162)
		tmp = t_2;
	elseif (z <= -1.02e-151)
		tmp = t_1;
	elseif (z <= 8.4e-100)
		tmp = Float64(t * Float64(y / a));
	elseif (z <= 4.9e+143)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x * (y / z);
	t_2 = t + (x - x);
	tmp = 0.0;
	if (z <= -1.1e+162)
		tmp = t_2;
	elseif (z <= -1.02e-151)
		tmp = t_1;
	elseif (z <= 8.4e-100)
		tmp = t * (y / a);
	elseif (z <= 4.9e+143)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(y / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t + N[(x - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.1e+162], t$95$2, If[LessEqual[z, -1.02e-151], t$95$1, If[LessEqual[z, 8.4e-100], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 4.9e+143], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1000000000000001e162 or 4.89999999999999986e143 < z

   1. Initial program 34.6%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 52.1

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

   5. Simplified 52.1%

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

      1. lift--.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 64.0

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

   7. Applied egg-rr 64.0%

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

   if -1.1000000000000001e162 < z < -1.0200000000000001e-151 or 8.40000000000000037e-100 < z < 4.89999999999999986e143

   1. Initial program 77.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 53.4%

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

   6. Taylor expanded in x around inf

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

      1. div-sub N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 33.1

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

   8. Simplified 33.1%

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

   9. Taylor expanded in y around inf

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

       1. lower-/.f64 31.4

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

   11. Simplified 31.4%

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

   if -1.0200000000000001e-151 < z < 8.40000000000000037e-100

   1. Initial program 91.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 93.7

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

   5. Simplified 93.7%

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

   6. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 38.6

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

   8. Simplified 38.6%

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

4. Final simplification 40.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+162}:\\ \;\;\;\;t + \left(x - x\right)\\ \mathbf{elif}\;z \leq -1.02 \cdot 10^{-151}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{-100}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;z \leq 4.9 \cdot 10^{+143}:\\ \;\;\;\;x \cdot \frac{y}{z}\\ \mathbf{else}:\\ \;\;\;\;t + \left(x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 86.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{if}\;a \leq -1.75 \cdot 10^{-123}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{-112}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ (- y z) (- a z)) x)))
   (if (<= a -1.75e-123)
     t_1
     (if (<= a 9.2e-112) (fma (- x t) (/ y z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), ((y - z) / (a - z)), x);
	double tmp;
	if (a <= -1.75e-123) {
		tmp = t_1;
	} else if (a <= 9.2e-112) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x)
	tmp = 0.0
	if (a <= -1.75e-123)
		tmp = t_1;
	elseif (a <= 9.2e-112)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.75e-123], t$95$1, If[LessEqual[a, 9.2e-112], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.7499999999999999e-123 or 9.19999999999999963e-112 < a

   1. Initial program 77.0%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 93.3

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

   4. Applied egg-rr 93.3%

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

   if -1.7499999999999999e-123 < a < 9.19999999999999963e-112

   1. Initial program 61.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 93.7%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 93.7

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

   8. Simplified 93.7%

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

Alternative 5: 53.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.8 \cdot 10^{-74}:\\ \;\;\;\;y \cdot \frac{x - t}{z}\\ \mathbf{elif}\;z \leq 7.2 \cdot 10^{+81}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t) (/ y z) t)))
   (if (<= z -1.1e+162)
     t_1
     (if (<= z -3.8e-74)
       (* y (/ (- x t) z))
       (if (<= z 7.2e+81) (fma y (/ t a) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-t, (y / z), t);
	double tmp;
	if (z <= -1.1e+162) {
		tmp = t_1;
	} else if (z <= -3.8e-74) {
		tmp = y * ((x - t) / z);
	} else if (z <= 7.2e+81) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -1.1e+162)
		tmp = t_1;
	elseif (z <= -3.8e-74)
		tmp = Float64(y * Float64(Float64(x - t) / z));
	elseif (z <= 7.2e+81)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.1e+162], t$95$1, If[LessEqual[z, -3.8e-74], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 7.2e+81], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.1000000000000001e162 or 7.20000000000000011e81 < z

   1. Initial program 35.1%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 83.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 77.7

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

   8. Simplified 77.7%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 63.4

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

   11. Simplified 63.4%

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

   if -1.1000000000000001e162 < z < -3.7999999999999996e-74

   1. Initial program 73.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 62.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 59.2

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

   8. Simplified 59.2%

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

   9. Taylor expanded in y around inf

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

       1. div-sub N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower--.f64 44.0

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

   11. Simplified 44.0%

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

   if -3.7999999999999996e-74 < z < 7.20000000000000011e81

   1. Initial program 90.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 77.3

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

   5. Simplified 77.3%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 64.0

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

   8. Simplified 64.0%

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

Alternative 6: 49.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;a \leq -9.8 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \frac{x - t}{z}\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\left(t - x\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ t a) x)))
   (if (<= a -9.8e-70)
     t_1
     (if (<= a 3.2e-135)
       (* y (/ (- x t) z))
       (if (<= a 9.2e+60) (/ (* (- t x) y) a) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (t / a), x);
	double tmp;
	if (a <= -9.8e-70) {
		tmp = t_1;
	} else if (a <= 3.2e-135) {
		tmp = y * ((x - t) / z);
	} else if (a <= 9.2e+60) {
		tmp = ((t - x) * y) / a;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(t / a), x)
	tmp = 0.0
	if (a <= -9.8e-70)
		tmp = t_1;
	elseif (a <= 3.2e-135)
		tmp = Float64(y * Float64(Float64(x - t) / z));
	elseif (a <= 9.2e+60)
		tmp = Float64(Float64(Float64(t - x) * y) / a);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -9.8e-70], t$95$1, If[LessEqual[a, 3.2e-135], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 9.2e+60], N[(N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision] / a), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.8000000000000001e-70 or 9.20000000000000068e60 < a

   1. Initial program 79.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 74.5

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

   5. Simplified 74.5%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 67.2

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

   8. Simplified 67.2%

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

   if -9.8000000000000001e-70 < a < 3.2e-135

   1. Initial program 63.2%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 92.5%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 92.5

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

   8. Simplified 92.5%

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

   9. Taylor expanded in y around inf

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

       1. div-sub N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower--.f64 58.3

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

   11. Simplified 58.3%

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

   if 3.2e-135 < a < 9.20000000000000068e60

   1. Initial program 69.8%

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

   3. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 54.8

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

   5. Simplified 54.8%

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

   6. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 45.6

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

   8. Simplified 45.6%

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

4. Final simplification 60.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -9.8 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \leq 3.2 \cdot 10^{-135}:\\ \;\;\;\;y \cdot \frac{x - t}{z}\\ \mathbf{elif}\;a \leq 9.2 \cdot 10^{+60}:\\ \;\;\;\;\frac{\left(t - x\right) \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -7.8e-116)
   (fma (- y z) (/ t (- a z)) x)
   (if (<= a 6e+23)
     (fma (- x t) (/ (- y a) z) t)
     (fma (- t x) (/ (- y z) a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -7.8e-116) {
		tmp = fma((y - z), (t / (a - z)), x);
	} else if (a <= 6e+23) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = fma((t - x), ((y - z) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -7.8e-116)
		tmp = fma(Float64(y - z), Float64(t / Float64(a - z)), x);
	elseif (a <= 6e+23)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	else
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -7.8e-116], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 6e+23], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -7.8000000000000001e-116

   1. Initial program 81.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 98.8

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

   4. Applied egg-rr 98.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 94.1

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

   6. Applied egg-rr 94.1%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 83.3

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

   9. Simplified 83.3%

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

   if -7.8000000000000001e-116 < a < 6.0000000000000002e23

   1. Initial program 63.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 80.8%

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

   if 6.0000000000000002e23 < a

   1. Initial program 75.2%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 94.7

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

   4. Applied egg-rr 94.7%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 85.3

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

   7. Simplified 85.3%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -7.8 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;a \leq 6 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 74.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -6.6 \cdot 10^{-116}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a - z}, x\right)\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -6.6e-116)
   (fma (- y z) (/ t (- a z)) x)
   (if (<= a 6.8e-34) (fma (- x t) (/ y z) t) (fma (- t x) (/ (- y z) a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -6.6e-116) {
		tmp = fma((y - z), (t / (a - z)), x);
	} else if (a <= 6.8e-34) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = fma((t - x), ((y - z) / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -6.6e-116)
		tmp = fma(Float64(y - z), Float64(t / Float64(a - z)), x);
	elseif (a <= 6.8e-34)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -6.6e-116], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 6.8e-34], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -6.60000000000000002e-116

   1. Initial program 81.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 98.8

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

   4. Applied egg-rr 98.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. div-inv N/A

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

      8. lift-/.f64 N/A

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

      9. associate-*l* N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-*.f64 94.1

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

   6. Applied egg-rr 94.1%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 83.3

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

   9. Simplified 83.3%

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

   if -6.60000000000000002e-116 < a < 6.8000000000000001e-34

   1. Initial program 63.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 84.3%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 83.0

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

   8. Simplified 83.0%

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

   if 6.8000000000000001e-34 < a

   1. Initial program 73.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 91.6

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

   4. Applied egg-rr 91.6%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 78.6

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

   7. Simplified 78.6%

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

Alternative 9: 73.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{if}\;a \leq -6.5 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.8 \cdot 10^{-34}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t x) (/ (- y z) a) x)))
   (if (<= a -6.5e-70) t_1 (if (<= a 6.8e-34) (fma (- x t) (/ y z) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t - x), ((y - z) / a), x);
	double tmp;
	if (a <= -6.5e-70) {
		tmp = t_1;
	} else if (a <= 6.8e-34) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t - x), Float64(Float64(y - z) / a), x)
	tmp = 0.0
	if (a <= -6.5e-70)
		tmp = t_1;
	elseif (a <= 6.8e-34)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -6.5e-70], t$95$1, If[LessEqual[a, 6.8e-34], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -6.5000000000000005e-70 or 6.8000000000000001e-34 < a

   1. Initial program 77.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 95.4

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

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 78.5

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

   7. Simplified 78.5%

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

   if -6.5000000000000005e-70 < a < 6.8000000000000001e-34

   1. Initial program 64.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 83.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 82.4

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

   8. Simplified 82.4%

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

Alternative 10: 69.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.9 \cdot 10^{-69}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 3.8 \cdot 10^{-53}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.9e-69)
   (fma y (/ (- t x) a) x)
   (if (<= a 3.8e-53) (fma (- x t) (/ y z) t) (fma (- t x) (/ y a) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.9e-69) {
		tmp = fma(y, ((t - x) / a), x);
	} else if (a <= 3.8e-53) {
		tmp = fma((x - t), (y / z), t);
	} else {
		tmp = fma((t - x), (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.9e-69)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	elseif (a <= 3.8e-53)
		tmp = fma(Float64(x - t), Float64(y / z), t);
	else
		tmp = fma(Float64(t - x), Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.9e-69], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[a, 3.8e-53], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if a < -1.8999999999999999e-69

   1. Initial program 81.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 73.6

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

   5. Simplified 73.6%

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

   if -1.8999999999999999e-69 < a < 3.7999999999999998e-53

   1. Initial program 64.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 85.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 84.4

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

   8. Simplified 84.4%

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

   if 3.7999999999999998e-53 < a

   1. Initial program 72.8%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 89.7

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

   4. Applied egg-rr 89.7%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 71.2

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

   7. Simplified 71.2%

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

Alternative 11: 62.4% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t) (/ y z) t)))
   (if (<= z -1.25e+153)
     t_1
     (if (<= z 4.7e+116) (fma (- t x) (/ y a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-t, (y / z), t);
	double tmp;
	if (z <= -1.25e+153) {
		tmp = t_1;
	} else if (z <= 4.7e+116) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -1.25e+153)
		tmp = t_1;
	elseif (z <= 4.7e+116)
		tmp = fma(Float64(t - x), Float64(y / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.25e+153], t$95$1, If[LessEqual[z, 4.7e+116], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000005e153 or 4.7000000000000003e116 < z

   1. Initial program 32.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 86.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 80.5

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

   8. Simplified 80.5%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 65.2

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

   11. Simplified 65.2%

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

   if -1.25000000000000005e153 < z < 4.7000000000000003e116

   1. Initial program 85.7%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lift-/.f64 N/A

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

      6. +-commutative N/A

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

      7. lift-/.f64 N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-/.f64 92.5

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

   4. Applied egg-rr 92.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 68.9

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

   7. Simplified 68.9%

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

Alternative 12: 61.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+153}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 4.7 \cdot 10^{+116}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- t) (/ y z) t)))
   (if (<= z -1.25e+153)
     t_1
     (if (<= z 4.7e+116) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(-t, (y / z), t);
	double tmp;
	if (z <= -1.25e+153) {
		tmp = t_1;
	} else if (z <= 4.7e+116) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(-t), Float64(y / z), t)
	tmp = 0.0
	if (z <= -1.25e+153)
		tmp = t_1;
	elseif (z <= 4.7e+116)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.25e+153], t$95$1, If[LessEqual[z, 4.7e+116], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.25000000000000005e153 or 4.7000000000000003e116 < z

   1. Initial program 32.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 86.6%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 80.5

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

   8. Simplified 80.5%

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

   9. Taylor expanded in x around 0

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

       1. mul-1-neg N/A

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

       2. lower-neg.f64 65.2

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

   11. Simplified 65.2%

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

   if -1.25000000000000005e153 < z < 4.7000000000000003e116

   1. Initial program 85.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 68.3

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

   5. Simplified 68.3%

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

Alternative 13: 49.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{if}\;a \leq -9.8 \cdot 10^{-70}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{-110}:\\ \;\;\;\;y \cdot \frac{x - t}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ t a) x)))
   (if (<= a -9.8e-70) t_1 (if (<= a 6.2e-110) (* y (/ (- x t) z)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (t / a), x);
	double tmp;
	if (a <= -9.8e-70) {
		tmp = t_1;
	} else if (a <= 6.2e-110) {
		tmp = y * ((x - t) / z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(t / a), x)
	tmp = 0.0
	if (a <= -9.8e-70)
		tmp = t_1;
	elseif (a <= 6.2e-110)
		tmp = Float64(y * Float64(Float64(x - t) / z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -9.8e-70], t$95$1, If[LessEqual[a, 6.2e-110], N[(y * N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -9.8000000000000001e-70 or 6.20000000000000014e-110 < a

   1. Initial program 76.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 69.7

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

   5. Simplified 69.7%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 58.3

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

   8. Simplified 58.3%

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

   if -9.8000000000000001e-70 < a < 6.20000000000000014e-110

   1. Initial program 63.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 91.9%

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

   6. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 91.9

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

   8. Simplified 91.9%

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

   9. Taylor expanded in y around inf

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

       1. div-sub N/A

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

       2. lower-*.f64 N/A

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

       3. lower-/.f64 N/A

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

       4. lower--.f64 57.8

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

   11. Simplified 57.8%

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

Alternative 14: 51.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(t, \frac{a}{z}, t\right)\\ \mathbf{if}\;z \leq -1.1 \cdot 10^{+172}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9.5 \cdot 10^{+89}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma t (/ a z) t)))
   (if (<= z -1.1e+172) t_1 (if (<= z 9.5e+89) (fma y (/ t a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(t, (a / z), t);
	double tmp;
	if (z <= -1.1e+172) {
		tmp = t_1;
	} else if (z <= 9.5e+89) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(t, Float64(a / z), t)
	tmp = 0.0
	if (z <= -1.1e+172)
		tmp = t_1;
	elseif (z <= 9.5e+89)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(a / z), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.1e+172], t$95$1, If[LessEqual[z, 9.5e+89], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1000000000000001e172 or 9.5000000000000003e89 < z

   1. Initial program 33.5%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

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

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Simplified 85.0%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

      4. div-sub N/A

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

      5. *-commutative N/A

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

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

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

      7. lower-fma.f64 N/A

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

      8. div-sub N/A

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

      9. lower-/.f64 N/A

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

      10. lower--.f64 N/A

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

      11. lower-neg.f64 65.2

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

   8. Simplified 65.2%

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

   9. Taylor expanded in x around 0

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

       1. +-commutative N/A

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

       2. *-commutative N/A

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

       3. associate-*r/ N/A

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

       4. lower-fma.f64 N/A

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

       5. lower-/.f64 59.4

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

   11. Simplified 59.4%

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

   if -1.1000000000000001e172 < z < 9.5000000000000003e89

   1. Initial program 85.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 67.0

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

   5. Simplified 67.0%

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

   6. Taylor expanded in t around inf

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

      1. lower-/.f64 55.4

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

   8. Simplified 55.4%

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

Alternative 15: 36.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t + \left(x - x\right)\\ \mathbf{if}\;z \leq -2 \cdot 10^{-34}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-9}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ t (- x x))))
   (if (<= z -2e-34) t_1 (if (<= z 1.45e-9) (* t (/ y a)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t + (x - x);
	double tmp;
	if (z <= -2e-34) {
		tmp = t_1;
	} else if (z <= 1.45e-9) {
		tmp = t * (y / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = t + (x - x)
    if (z <= (-2d-34)) then
        tmp = t_1
    else if (z <= 1.45d-9) then
        tmp = t * (y / a)
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t + (x - x);
	double tmp;
	if (z <= -2e-34) {
		tmp = t_1;
	} else if (z <= 1.45e-9) {
		tmp = t * (y / a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t + (x - x)
	tmp = 0
	if z <= -2e-34:
		tmp = t_1
	elif z <= 1.45e-9:
		tmp = t * (y / a)
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t + Float64(x - x))
	tmp = 0.0
	if (z <= -2e-34)
		tmp = t_1;
	elseif (z <= 1.45e-9)
		tmp = Float64(t * Float64(y / a));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t + (x - x);
	tmp = 0.0;
	if (z <= -2e-34)
		tmp = t_1;
	elseif (z <= 1.45e-9)
		tmp = t * (y / a);
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t + N[(x - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -2e-34], t$95$1, If[LessEqual[z, 1.45e-9], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.99999999999999986e-34 or 1.44999999999999996e-9 < z

   1. Initial program 56.3%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 29.3

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

   5. Simplified 29.3%

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

      1. lift--.f64 N/A

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

      2. +-commutative N/A

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

      3. lift--.f64 N/A

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

      4. associate-+l- N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 37.5

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

   7. Applied egg-rr 37.5%

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

   if -1.99999999999999986e-34 < z < 1.44999999999999996e-9

   1. Initial program 90.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 83.4

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

   5. Simplified 83.4%

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

   6. Taylor expanded in t around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 32.2

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

   8. Simplified 32.2%

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

4. Final simplification 34.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2 \cdot 10^{-34}:\\ \;\;\;\;t + \left(x - x\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-9}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{else}:\\ \;\;\;\;t + \left(x - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 25.5% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 72.5%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 16.7

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

5. Simplified 16.7%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
6. Step-by-step derivation

   1. lift--.f64 N/A

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\left(t - x\right) + x} \]

   3. lift--.f64 N/A

      \[\leadsto \color{blue}{\left(t - x\right)} + x \]

   4. associate-+l- N/A

      \[\leadsto \color{blue}{t - \left(x - x\right)} \]

   5. lower--.f64 N/A

      \[\leadsto \color{blue}{t - \left(x - x\right)} \]

   6. lower--.f64 21.0

      \[\leadsto t - \color{blue}{\left(x - x\right)} \]

7. Applied egg-rr 21.0%

   \[\leadsto \color{blue}{t - \left(x - x\right)} \]

8. Final simplification 21.0%

   \[\leadsto t + \left(x - x\right) \]
9. Add Preprocessing

Alternative 17: 19.8% accurate, 4.1× speedup

Math

\[\begin{array}{l} \\ x + \left(t - x\right) \end{array} \]

FPCore

(FPCore (x y z t a) :precision binary64 (+ x (- t x)))

C

double code(double x, double y, double z, double t, double a) {
	return x + (t - x);
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x + (t - x)
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	return x + (t - x);
}

Python

def code(x, y, z, t, a):
	return x + (t - x)

Julia

function code(x, y, z, t, a)
	return Float64(x + Float64(t - x))
end

MATLAB

function tmp = code(x, y, z, t, a)
	tmp = x + (t - x);
end

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 72.5%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
4. Step-by-step derivation

   1. lower--.f64 16.7

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

5. Simplified 16.7%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
6. Add Preprocessing

Alternative 18: 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 72.5%

   \[x + \frac{\left(y - z\right) \cdot \left(t - x\right)}{a - z} \]
2. Add Preprocessing

3. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
4. Step-by-step derivation

   1. lower--.f64 16.7

      \[\leadsto x + \color{blue}{\left(t - x\right)} \]

5. Simplified 16.7%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]

6. Taylor expanded in t around 0

   \[\leadsto x + \color{blue}{-1 \cdot x} \]
7. Step-by-step derivation

   1. mul-1-neg N/A

      \[\leadsto x + \color{blue}{\left(\mathsf{neg}\left(x\right)\right)} \]

   2. lower-neg.f64 2.6

      \[\leadsto x + \color{blue}{\left(-x\right)} \]

8. Simplified 2.6%

   \[\leadsto x + \color{blue}{\left(-x\right)} \]
9. Step-by-step derivation

   1. unsub-neg N/A

      \[\leadsto \color{blue}{x - x} \]

   2. +-inverses 2.6

      \[\leadsto \color{blue}{0} \]

10. Applied egg-rr 2.6%

    \[\leadsto \color{blue}{0} \]
11. Add Preprocessing

Developer Target 1: 84.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t - \frac{y}{z} \cdot \left(t - x\right)\\ \mathbf{if}\;z < -1.2536131056095036 \cdot 10^{+188}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z < 4.446702369113811 \cdot 10^{+64}:\\ \;\;\;\;x + \frac{y - z}{\frac{a - z}{t - x}}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- t (* (/ y z) (- t x)))))
   (if (< z -1.2536131056095036e+188)
     t_1
     (if (< z 4.446702369113811e+64)
       (+ x (/ (- y z) (/ (- a z) (- t x))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - 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 = t - ((y / z) * (t - x))
    if (z < (-1.2536131056095036d+188)) then
        tmp = t_1
    else if (z < 4.446702369113811d+64) then
        tmp = x + ((y - z) / ((a - z) / (t - 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 = t - ((y / z) * (t - x));
	double tmp;
	if (z < -1.2536131056095036e+188) {
		tmp = t_1;
	} else if (z < 4.446702369113811e+64) {
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t - ((y / z) * (t - x))
	tmp = 0
	if z < -1.2536131056095036e+188:
		tmp = t_1
	elif z < 4.446702369113811e+64:
		tmp = x + ((y - z) / ((a - z) / (t - x)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t - Float64(Float64(y / z) * Float64(t - x)))
	tmp = 0.0
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = Float64(x + Float64(Float64(y - z) / Float64(Float64(a - z) / Float64(t - x))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t - ((y / z) * (t - x));
	tmp = 0.0;
	if (z < -1.2536131056095036e+188)
		tmp = t_1;
	elseif (z < 4.446702369113811e+64)
		tmp = x + ((y - z) / ((a - z) / (t - x)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t - N[(N[(y / z), $MachinePrecision] * N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[z, -1.2536131056095036e+188], t$95$1, If[Less[z, 4.446702369113811e+64], N[(x + N[(N[(y - z), $MachinePrecision] / N[(N[(a - z), $MachinePrecision] / N[(t - x), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.Axis.Types:invLinMap from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< z -125361310560950360000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- t (* (/ y z) (- t x))) (if (< z 44467023691138110000000000000000000000000000000000000000000000000) (+ x (/ (- y z) (/ (- a z) (- t x)))) (- t (* (/ y z) (- t x))))))

  (+ x (/ (* (- y z) (- t x)) (- a z))))