Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B

Percentage Accurate: 98.3% → 99.0%

Time: 9.8s

Alternatives: 11

Speedup: 0.5×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.3% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{+245}:\\ \;\;\;\;x + t\_1 \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t - a}, t - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 4e+245) (+ x (* t_1 y)) (fma (/ y (- t a)) (- t z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double tmp;
	if (t_1 <= 4e+245) {
		tmp = x + (t_1 * y);
	} else {
		tmp = fma((y / (t - a)), (t - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= 4e+245)
		tmp = Float64(x + Float64(t_1 * y));
	else
		tmp = fma(Float64(y / Float64(t - a)), Float64(t - z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 4e+245], N[(x + N[(t$95$1 * y), $MachinePrecision]), $MachinePrecision], N[(N[(y / N[(t - a), $MachinePrecision]), $MachinePrecision] * N[(t - z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 4.00000000000000018e245

   1. Initial program 99.1%

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

   if 4.00000000000000018e245 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 47.3%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. frac-2neg N/A

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

      4. frac-2neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. clear-num N/A

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

      11. un-div-inv N/A

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

      12. frac-2neg N/A

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

      13. associate-/r/ N/A

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

      14. lower-fma.f64 N/A

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

   4. Applied egg-rr 99.7%

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

4. Final simplification 99.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 4 \cdot 10^{+245}:\\ \;\;\;\;x + \frac{z - t}{a - t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t - a}, t - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 83.6% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \frac{z \cdot y}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+88}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 1.002:\\ \;\;\;\;x + y\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (/ (* z y) (- a t))))
   (if (<= t_1 -1e+88)
     t_2
     (if (<= t_1 5e-12)
       (fma z (/ y a) x)
       (if (<= t_1 1.002)
         (+ x y)
         (if (<= t_1 2e+153) (fma y (/ z a) x) t_2))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = (z * y) / (a - t);
	double tmp;
	if (t_1 <= -1e+88) {
		tmp = t_2;
	} else if (t_1 <= 5e-12) {
		tmp = fma(z, (y / a), x);
	} else if (t_1 <= 1.002) {
		tmp = x + y;
	} else if (t_1 <= 2e+153) {
		tmp = fma(y, (z / a), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(Float64(z * y) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -1e+88)
		tmp = t_2;
	elseif (t_1 <= 5e-12)
		tmp = fma(z, Float64(y / a), x);
	elseif (t_1 <= 1.002)
		tmp = Float64(x + y);
	elseif (t_1 <= 2e+153)
		tmp = fma(y, Float64(z / a), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+88], t$95$2, If[LessEqual[t$95$1, 5e-12], N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 1.002], N[(x + y), $MachinePrecision], If[LessEqual[t$95$1, 2e+153], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -9.99999999999999959e87 or 2e153 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 83.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 73.0

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

   5. Simplified 73.0%

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

      1. lift--.f64 N/A

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

      2. div-inv N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 82.7

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

   7. Applied egg-rr 82.7%

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

   if -9.99999999999999959e87 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999997e-12

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 79.1

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

   5. Simplified 79.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 84.1

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

   7. Applied egg-rr 84.1%

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

   if 4.9999999999999997e-12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.3

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

   5. Simplified 97.3%

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

   if 1.002 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2e153

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 84.0

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

   5. Simplified 84.0%

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

4. Final simplification 88.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1 \cdot 10^{+88}:\\ \;\;\;\;\frac{z \cdot y}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 1.002:\\ \;\;\;\;x + y\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{+153}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 88.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \frac{z \cdot y}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+88}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (/ (* z y) (- a t))))
   (if (<= t_1 -1e+88)
     t_2
     (if (<= t_1 2e-5)
       (fma y (/ (- z t) a) x)
       (if (<= t_1 2e+54) (fma y (- 1.0 (/ z t)) x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = (z * y) / (a - t);
	double tmp;
	if (t_1 <= -1e+88) {
		tmp = t_2;
	} else if (t_1 <= 2e-5) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 2e+54) {
		tmp = fma(y, (1.0 - (z / t)), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(Float64(z * y) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -1e+88)
		tmp = t_2;
	elseif (t_1 <= 2e-5)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t_1 <= 2e+54)
		tmp = fma(y, Float64(1.0 - Float64(z / t)), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+88], t$95$2, If[LessEqual[t$95$1, 2e-5], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+54], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -9.99999999999999959e87 or 2.0000000000000002e54 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 86.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 73.7

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

   5. Simplified 73.7%

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

      1. lift--.f64 N/A

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

      2. div-inv N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 79.3

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

   7. Applied egg-rr 79.3%

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

   if -9.99999999999999959e87 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2.00000000000000016e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 96.6

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

   5. Simplified 96.6%

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

   if 2.00000000000000016e-5 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2.0000000000000002e54

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 97.0

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

   5. Simplified 97.0%

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

4. Final simplification 93.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1 \cdot 10^{+88}:\\ \;\;\;\;\frac{z \cdot y}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 83.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \frac{z \cdot y}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+88}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (/ (* z y) (- a t))))
   (if (<= t_1 -1e+88)
     t_2
     (if (<= t_1 2e-5)
       (fma y (/ z a) x)
       (if (<= t_1 2e+54) (fma y (- 1.0 (/ z t)) x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = (z * y) / (a - t);
	double tmp;
	if (t_1 <= -1e+88) {
		tmp = t_2;
	} else if (t_1 <= 2e-5) {
		tmp = fma(y, (z / a), x);
	} else if (t_1 <= 2e+54) {
		tmp = fma(y, (1.0 - (z / t)), x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(Float64(z * y) / Float64(a - t))
	tmp = 0.0
	if (t_1 <= -1e+88)
		tmp = t_2;
	elseif (t_1 <= 2e-5)
		tmp = fma(y, Float64(z / a), x);
	elseif (t_1 <= 2e+54)
		tmp = fma(y, Float64(1.0 - Float64(z / t)), x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z * y), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+88], t$95$2, If[LessEqual[t$95$1, 2e-5], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+54], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -9.99999999999999959e87 or 2.0000000000000002e54 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 86.7%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 73.7

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

   5. Simplified 73.7%

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

      1. lift--.f64 N/A

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

      2. div-inv N/A

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

      3. associate-*r* N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-/.f64 N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 79.3

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

   7. Applied egg-rr 79.3%

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

   if -9.99999999999999959e87 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2.00000000000000016e-5

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 83.4

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

   5. Simplified 83.4%

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

   if 2.00000000000000016e-5 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2.0000000000000002e54

   1. Initial program 100.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. div-sub N/A

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

      6. sub-neg N/A

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

      7. *-inverses N/A

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

      8. metadata-eval N/A

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

      9. +-commutative N/A

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

      10. distribute-neg-in N/A

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

      11. metadata-eval N/A

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

      12. mul-1-neg N/A

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

      13. lower-fma.f64 N/A

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

      14. mul-1-neg N/A

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

      15. unsub-neg N/A

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

      16. lower--.f64 N/A

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

      17. lower-/.f64 97.0

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

   5. Simplified 97.0%

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

4. Final simplification 88.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1 \cdot 10^{+88}:\\ \;\;\;\;\frac{z \cdot y}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{+54}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot y}{a - t}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 81.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ t_2 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+88}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1.002:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y a) x)) (t_2 (/ (- z t) (- a t))))
   (if (<= t_2 -1e+88)
     (* y (/ z (- a t)))
     (if (<= t_2 5e-12) t_1 (if (<= t_2 1.002) (+ x y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / a), x);
	double t_2 = (z - t) / (a - t);
	double tmp;
	if (t_2 <= -1e+88) {
		tmp = y * (z / (a - t));
	} else if (t_2 <= 5e-12) {
		tmp = t_1;
	} else if (t_2 <= 1.002) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / a), x)
	t_2 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_2 <= -1e+88)
		tmp = Float64(y * Float64(z / Float64(a - t)));
	elseif (t_2 <= 5e-12)
		tmp = t_1;
	elseif (t_2 <= 1.002)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+88], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e-12], t$95$1, If[LessEqual[t$95$2, 1.002], N[(x + y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -9.99999999999999959e87

   1. Initial program 91.0%

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

   3. Taylor expanded in z around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 77.2

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

   5. Simplified 77.2%

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

   if -9.99999999999999959e87 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999997e-12 or 1.002 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 96.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 74.6

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

   5. Simplified 74.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 80.1

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

   7. Applied egg-rr 80.1%

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

   if 4.9999999999999997e-12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.3

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

   5. Simplified 97.3%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 86.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1 \cdot 10^{+88}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 1.002:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 80.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ t_2 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 1.002:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma z (/ y a) x)) (t_2 (/ (- z t) (- a t))))
   (if (<= t_2 -5e+211)
     (/ (* z (- y)) t)
     (if (<= t_2 5e-12) t_1 (if (<= t_2 1.002) (+ x y) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(z, (y / a), x);
	double t_2 = (z - t) / (a - t);
	double tmp;
	if (t_2 <= -5e+211) {
		tmp = (z * -y) / t;
	} else if (t_2 <= 5e-12) {
		tmp = t_1;
	} else if (t_2 <= 1.002) {
		tmp = x + y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(z, Float64(y / a), x)
	t_2 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_2 <= -5e+211)
		tmp = Float64(Float64(z * Float64(-y)) / t);
	elseif (t_2 <= 5e-12)
		tmp = t_1;
	elseif (t_2 <= 1.002)
		tmp = Float64(x + y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+211], N[(N[(z * (-y)), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$2, 5e-12], t$95$1, If[LessEqual[t$95$2, 1.002], N[(x + y), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -4.9999999999999995e211

   1. Initial program 85.5%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. frac-2neg N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. associate--r+ N/A

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

      13. neg-sub0 N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 N/A

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

      16. neg-sub0 N/A

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

      17. lift--.f64 N/A

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

      18. sub-neg N/A

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

      19. +-commutative N/A

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

      20. associate--r+ N/A

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

      21. neg-sub0 N/A

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

      22. remove-double-neg N/A

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

      23. lower--.f64 85.5

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

   4. Applied egg-rr 85.5%

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

   5. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. mul-1-neg N/A

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

      8. lower-fma.f64 N/A

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

      9. mul-1-neg N/A

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

      10. unsub-neg N/A

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

      11. lower--.f64 N/A

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

      12. lower-/.f64 68.5

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

   7. Simplified 68.5%

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

   8. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \frac{y \cdot z}{t}} \]
   9. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-*.f64 75.7

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

   10. Simplified 75.7%

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

   if -4.9999999999999995e211 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999997e-12 or 1.002 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 96.6%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 73.2

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

   5. Simplified 73.2%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 78.5

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

   7. Applied egg-rr 78.5%

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

   if 4.9999999999999997e-12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.3

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

   5. Simplified 97.3%

      \[\leadsto \color{blue}{y + x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 85.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -5 \cdot 10^{+211}:\\ \;\;\;\;\frac{z \cdot \left(-y\right)}{t}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 1.002:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 80.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-12}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.002:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma z (/ y a) x)))
   (if (<= t_1 5e-12) t_2 (if (<= t_1 1.002) (+ x y) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma(z, (y / a), x);
	double tmp;
	if (t_1 <= 5e-12) {
		tmp = t_2;
	} else if (t_1 <= 1.002) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(z, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 5e-12)
		tmp = t_2;
	elseif (t_1 <= 1.002)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-12], t$95$2, If[LessEqual[t$95$1, 1.002], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 4.9999999999999997e-12 or 1.002 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.7%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 70.5

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

   5. Simplified 70.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 75.3

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

   7. Applied egg-rr 75.3%

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

   if 4.9999999999999997e-12 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.002

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.3

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

   5. Simplified 97.3%

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

4. Final simplification 83.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 1.002:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{y}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 2 \cdot 10^{-5}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.002:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (fma y (/ z a) x)))
   (if (<= t_1 2e-5) t_2 (if (<= t_1 1.002) (+ x y) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = fma(y, (z / a), x);
	double tmp;
	if (t_1 <= 2e-5) {
		tmp = t_2;
	} else if (t_1 <= 1.002) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = fma(y, Float64(z / a), x)
	tmp = 0.0
	if (t_1 <= 2e-5)
		tmp = t_2;
	elseif (t_1 <= 1.002)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 2e-5], t$95$2, If[LessEqual[t$95$1, 1.002], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 2.00000000000000016e-5 or 1.002 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 73.7

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

   5. Simplified 73.7%

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

   if 2.00000000000000016e-5 < (/.f64 (-.f64 z t) (-.f64 a t)) < 1.002

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.1

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

   5. Simplified 98.1%

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

4. Final simplification 82.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 1.002:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 64.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := z \cdot \frac{y}{a}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+14}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+54}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))) (t_2 (* z (/ y a))))
   (if (<= t_1 -1e+14) t_2 (if (<= t_1 2e+54) (+ x y) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (a - t);
	double t_2 = z * (y / a);
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = t_2;
	} else if (t_1 <= 2e+54) {
		tmp = x + y;
	} 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 = (z - t) / (a - t)
    t_2 = z * (y / a)
    if (t_1 <= (-1d+14)) then
        tmp = t_2
    else if (t_1 <= 2d+54) then
        tmp = x + y
    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 = (z - t) / (a - t);
	double t_2 = z * (y / a);
	double tmp;
	if (t_1 <= -1e+14) {
		tmp = t_2;
	} else if (t_1 <= 2e+54) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = (z - t) / (a - t)
	t_2 = z * (y / a)
	tmp = 0
	if t_1 <= -1e+14:
		tmp = t_2
	elif t_1 <= 2e+54:
		tmp = x + y
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(a - t))
	t_2 = Float64(z * Float64(y / a))
	tmp = 0.0
	if (t_1 <= -1e+14)
		tmp = t_2;
	elseif (t_1 <= 2e+54)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = (z - t) / (a - t);
	t_2 = z * (y / a);
	tmp = 0.0;
	if (t_1 <= -1e+14)
		tmp = t_2;
	elseif (t_1 <= 2e+54)
		tmp = x + y;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+14], t$95$2, If[LessEqual[t$95$1, 2e+54], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1e14 or 2.0000000000000002e54 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 89.5%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 61.8

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

   5. Simplified 61.8%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 44.2

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

   8. Simplified 44.2%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 45.8

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

   10. Applied egg-rr 45.8%

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

   if -1e14 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2.0000000000000002e54

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 75.0

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

   5. Simplified 75.0%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{a - t} \leq -1 \cdot 10^{+14}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \mathbf{elif}\;\frac{z - t}{a - t} \leq 2 \cdot 10^{+54}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 95.7% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 97.3%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. frac-2neg N/A

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

   4. frac-2neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. lift-/.f64 N/A

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

   10. clear-num N/A

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

   11. un-div-inv N/A

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

   12. frac-2neg N/A

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

   13. associate-/r/ N/A

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

   14. lower-fma.f64 N/A

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

4. Applied egg-rr 96.7%

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

Alternative 11: 60.2% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + y), $MachinePrecision]

Derivation

1. Initial program 97.3%

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

3. Taylor expanded in t around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 62.5

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

5. Simplified 62.5%

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

6. Final simplification 62.5%

   \[\leadsto x + y \]
7. Add Preprocessing

Developer Target 1: 99.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;y < -8.508084860551241 \cdot 10^{-17}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y < 2.894426862792089 \cdot 10^{-49}:\\ \;\;\;\;x + \left(y \cdot \left(z - t\right)\right) \cdot \frac{1}{a - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* y (/ (- z t) (- a t))))))
   (if (< y -8.508084860551241e-17)
     t_1
     (if (< y 2.894426862792089e-49)
       (+ x (* (* y (- z t)) (/ 1.0 (- a t))))
       t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x + (y * ((z - t) / (a - t)))
    if (y < (-8.508084860551241d-17)) then
        tmp = t_1
    else if (y < 2.894426862792089d-49) then
        tmp = x + ((y * (z - t)) * (1.0d0 / (a - t)))
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = x + (y * ((z - t) / (a - t)));
	double tmp;
	if (y < -8.508084860551241e-17) {
		tmp = t_1;
	} else if (y < 2.894426862792089e-49) {
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = x + (y * ((z - t) / (a - t)))
	tmp = 0
	if y < -8.508084860551241e-17:
		tmp = t_1
	elif y < 2.894426862792089e-49:
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(y * Float64(Float64(z - t) / Float64(a - t))))
	tmp = 0.0
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = Float64(x + Float64(Float64(y * Float64(z - t)) * Float64(1.0 / Float64(a - t))));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = x + (y * ((z - t) / (a - t)));
	tmp = 0.0;
	if (y < -8.508084860551241e-17)
		tmp = t_1;
	elseif (y < 2.894426862792089e-49)
		tmp = x + ((y * (z - t)) * (1.0 / (a - t)));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Less[y, -8.508084860551241e-17], t$95$1, If[Less[y, 2.894426862792089e-49], N[(x + N[(N[(y * N[(z - t), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Reproduce

herbie shell --seed 2024207 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisLine from plot-0.2.3.4, B"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< y -8508084860551241/100000000000000000000000000000000) (+ x (* y (/ (- z t) (- a t)))) (if (< y 2894426862792089/10000000000000000000000000000000000000000000000000000000000000000) (+ x (* (* y (- z t)) (/ 1 (- a t)))) (+ x (* y (/ (- z t) (- a t)))))))

  (+ x (* y (/ (- z t) (- a t)))))