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

Percentage Accurate: 98.5% → 99.1%

Time: 40.8s

Alternatives: 10

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: 98.5% 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: 99.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq 10^{+292}:\\ \;\;\;\;t\_1 + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t - a}, t - z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ (- z t) (- a t)))))
   (if (<= t_1 1e+292) (+ t_1 x) (fma (/ y (- t a)) (- t z) x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * ((z - t) / (a - t));
	double tmp;
	if (t_1 <= 1e+292) {
		tmp = t_1 + x;
	} else {
		tmp = fma((y / (t - a)), (t - z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(Float64(z - t) / Float64(a - t)))
	tmp = 0.0
	if (t_1 <= 1e+292)
		tmp = Float64(t_1 + x);
	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[(y * N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e+292], N[(t$95$1 + x), $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 y (/.f64 (-.f64 z t) (-.f64 a t))) < 1e292

   1. Initial program 99.9%

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

   if 1e292 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 a t)))

   1. Initial program 78.3%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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. lift-/.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. +-commutative N/A

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

      6. lift-*.f64 N/A

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

      7. lift-/.f64 N/A

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

      8. clear-num N/A

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

      9. un-div-inv N/A

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

      10. frac-2neg N/A

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

      11. associate-/r/ N/A

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

      12. 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 rewrites 100.0%

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

4. Final simplification 99.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \cdot \frac{z - t}{a - t} \leq 10^{+292}:\\ \;\;\;\;y \cdot \frac{z - t}{a - t} + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{t - a}, t - z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 87.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+116}:\\ \;\;\;\;\frac{y \cdot z}{a - t}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z - t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -5e+116)
     (/ (* y z) (- a t))
     (if (<= t_1 0.002)
       (fma y (/ (- z t) a) x)
       (if (<= t_1 2e+15) (fma y (- 1.0 (/ z t)) x) (+ x (/ (* y z) a)))))))

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 <= -5e+116) {
		tmp = (y * z) / (a - t);
	} else if (t_1 <= 0.002) {
		tmp = fma(y, ((z - t) / a), x);
	} else if (t_1 <= 2e+15) {
		tmp = fma(y, (1.0 - (z / t)), x);
	} else {
		tmp = x + ((y * z) / a);
	}
	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 <= -5e+116)
		tmp = Float64(Float64(y * z) / Float64(a - t));
	elseif (t_1 <= 0.002)
		tmp = fma(y, Float64(Float64(z - t) / a), x);
	elseif (t_1 <= 2e+15)
		tmp = fma(y, Float64(1.0 - Float64(z / t)), x);
	else
		tmp = Float64(x + Float64(Float64(y * z) / a));
	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, -5e+116], N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(y * N[(N[(z - t), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+15], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -5.00000000000000025e116

   1. Initial program 87.5%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 87.5

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

   4. Applied rewrites 87.5%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 87.5

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 78.9

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

   9. Applied rewrites 78.9%

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

   if -5.00000000000000025e116 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2e-3

   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 91.7

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

   5. Applied rewrites 91.7%

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

   if 2e-3 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2e15

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 2e15 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.1%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

Alternative 3: 83.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+116}:\\ \;\;\;\;\frac{y \cdot z}{a - t}\\ \mathbf{elif}\;t\_1 \leq 0.002:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{z}{t}, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -5e+116)
     (/ (* y z) (- a t))
     (if (<= t_1 0.002)
       (fma y (/ z a) x)
       (if (<= t_1 2e+15) (fma y (- 1.0 (/ z t)) x) (+ x (/ (* y z) a)))))))

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 <= -5e+116) {
		tmp = (y * z) / (a - t);
	} else if (t_1 <= 0.002) {
		tmp = fma(y, (z / a), x);
	} else if (t_1 <= 2e+15) {
		tmp = fma(y, (1.0 - (z / t)), x);
	} else {
		tmp = x + ((y * z) / a);
	}
	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 <= -5e+116)
		tmp = Float64(Float64(y * z) / Float64(a - t));
	elseif (t_1 <= 0.002)
		tmp = fma(y, Float64(z / a), x);
	elseif (t_1 <= 2e+15)
		tmp = fma(y, Float64(1.0 - Float64(z / t)), x);
	else
		tmp = Float64(x + Float64(Float64(y * z) / a));
	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, -5e+116], N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 0.002], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+15], N[(y * N[(1.0 - N[(z / t), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -5.00000000000000025e116

   1. Initial program 87.5%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 87.5

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

   4. Applied rewrites 87.5%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 87.5

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 78.9

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

   9. Applied rewrites 78.9%

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

   if -5.00000000000000025e116 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2e-3

   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 83.7

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

   5. Applied rewrites 83.7%

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

   if 2e-3 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2e15

   1. Initial program 99.9%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 99.3

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

   5. Applied rewrites 99.3%

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

   if 2e15 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.1%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

Alternative 4: 82.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+116}:\\ \;\;\;\;\frac{y \cdot z}{a - t}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -5e+116)
     (/ (* y z) (- a t))
     (if (<= t_1 4e-8)
       (fma y (/ z a) x)
       (if (<= t_1 2e+15) (+ y x) (+ x (/ (* y z) a)))))))

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 <= -5e+116) {
		tmp = (y * z) / (a - t);
	} else if (t_1 <= 4e-8) {
		tmp = fma(y, (z / a), x);
	} else if (t_1 <= 2e+15) {
		tmp = y + x;
	} else {
		tmp = x + ((y * z) / a);
	}
	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 <= -5e+116)
		tmp = Float64(Float64(y * z) / Float64(a - t));
	elseif (t_1 <= 4e-8)
		tmp = fma(y, Float64(z / a), x);
	elseif (t_1 <= 2e+15)
		tmp = Float64(y + x);
	else
		tmp = Float64(x + Float64(Float64(y * z) / a));
	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, -5e+116], N[(N[(y * z), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-8], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+15], N[(y + x), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -5.00000000000000025e116

   1. Initial program 87.5%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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. lift--.f64 N/A

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

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. lower-fma.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lower-*.f64 N/A

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

      10. distribute-neg-frac2 N/A

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

      11. lower-/.f64 N/A

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

      12. neg-sub0 N/A

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

      13. lift--.f64 N/A

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

      14. sub-neg N/A

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

      15. +-commutative N/A

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

      16. associate--r+ N/A

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

      17. neg-sub0 N/A

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

      18. remove-double-neg N/A

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

      19. lower--.f64 87.5

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

   4. Applied rewrites 87.5%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. lift-fma.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-fma.f64 N/A

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

      9. associate-+l+ N/A

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

      10. lower-fma.f64 N/A

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

      11. lift-*.f64 N/A

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

      12. lower-fma.f64 87.5

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

   6. Applied rewrites 87.5%

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

   7. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 78.9

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

   9. Applied rewrites 78.9%

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

   if -5.00000000000000025e116 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.0000000000000001e-8

   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. Applied rewrites 83.4%

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

   if 4.0000000000000001e-8 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2e15

   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 97.5

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

   5. Applied rewrites 97.5%

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

   if 2e15 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.1%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

Alternative 5: 82.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \mathbf{elif}\;t\_1 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+15}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y \cdot z}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- a t))))
   (if (<= t_1 -1e+25)
     (* y (/ z (- a t)))
     (if (<= t_1 4e-8)
       (fma y (/ z a) x)
       (if (<= t_1 2e+15) (+ y x) (+ x (/ (* y z) a)))))))

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 <= -1e+25) {
		tmp = y * (z / (a - t));
	} else if (t_1 <= 4e-8) {
		tmp = fma(y, (z / a), x);
	} else if (t_1 <= 2e+15) {
		tmp = y + x;
	} else {
		tmp = x + ((y * z) / a);
	}
	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 <= -1e+25)
		tmp = Float64(y * Float64(z / Float64(a - t)));
	elseif (t_1 <= 4e-8)
		tmp = fma(y, Float64(z / a), x);
	elseif (t_1 <= 2e+15)
		tmp = Float64(y + x);
	else
		tmp = Float64(x + Float64(Float64(y * z) / a));
	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, -1e+25], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 4e-8], N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 2e+15], N[(y + x), $MachinePrecision], N[(x + N[(N[(y * z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.00000000000000009e25

   1. Initial program 92.6%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 65.3

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

   5. Applied rewrites 65.3%

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

   if -1.00000000000000009e25 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.0000000000000001e-8

   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 88.0

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

   5. Applied rewrites 88.0%

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

   if 4.0000000000000001e-8 < (/.f64 (-.f64 z t) (-.f64 a t)) < 2e15

   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 97.5

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

   5. Applied rewrites 97.5%

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

   if 2e15 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 95.1%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 75.8

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

   5. Applied rewrites 75.8%

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

Alternative 6: 82.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ t_2 := \frac{z - t}{a - t}\\ \mathbf{if}\;t\_2 \leq -1 \cdot 10^{+25}:\\ \;\;\;\;y \cdot \frac{z}{a - t}\\ \mathbf{elif}\;t\_2 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+17}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (/ z a) x)) (t_2 (/ (- z t) (- a t))))
   (if (<= t_2 -1e+25)
     (* y (/ z (- a t)))
     (if (<= t_2 4e-8) t_1 (if (<= t_2 5e+17) (+ y x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (z / a), x);
	double t_2 = (z - t) / (a - t);
	double tmp;
	if (t_2 <= -1e+25) {
		tmp = y * (z / (a - t));
	} else if (t_2 <= 4e-8) {
		tmp = t_1;
	} else if (t_2 <= 5e+17) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(z / a), x)
	t_2 = Float64(Float64(z - t) / Float64(a - t))
	tmp = 0.0
	if (t_2 <= -1e+25)
		tmp = Float64(y * Float64(z / Float64(a - t)));
	elseif (t_2 <= 4e-8)
		tmp = t_1;
	elseif (t_2 <= 5e+17)
		tmp = Float64(y + x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] / N[(a - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -1e+25], N[(y * N[(z / N[(a - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 4e-8], t$95$1, If[LessEqual[t$95$2, 5e+17], N[(y + x), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < -1.00000000000000009e25

   1. Initial program 92.6%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 65.3

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

   5. Applied rewrites 65.3%

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

   if -1.00000000000000009e25 < (/.f64 (-.f64 z t) (-.f64 a t)) < 4.0000000000000001e-8 or 5e17 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 98.4%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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. Applied rewrites 84.0%

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

   if 4.0000000000000001e-8 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5e17

   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 96.5

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

   5. Applied rewrites 96.5%

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

Alternative 7: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{a - t}\\ t_2 := \mathsf{fma}\left(y, \frac{z}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 4 \cdot 10^{-8}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+17}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

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 4e-8) t_2 (if (<= t_1 5e+17) (+ y 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 = fma(y, (z / a), x);
	double tmp;
	if (t_1 <= 4e-8) {
		tmp = t_2;
	} else if (t_1 <= 5e+17) {
		tmp = y + 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 = fma(y, Float64(z / a), x)
	tmp = 0.0
	if (t_1 <= 4e-8)
		tmp = t_2;
	elseif (t_1 <= 5e+17)
		tmp = Float64(y + 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[(y * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 4e-8], t$95$2, If[LessEqual[t$95$1, 5e+17], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 a t)) < 4.0000000000000001e-8 or 5e17 < (/.f64 (-.f64 z t) (-.f64 a t))

   1. Initial program 97.0%

      \[x + y \cdot \frac{z - t}{a - t} \]
   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 76.7

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

   5. Applied rewrites 76.7%

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

   if 4.0000000000000001e-8 < (/.f64 (-.f64 z t) (-.f64 a t)) < 5e17

   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 96.5

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

   5. Applied rewrites 96.5%

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

Alternative 8: 96.0% 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 98.0%

   \[x + y \cdot \frac{z - t}{a - t} \]
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. lift-/.f64 N/A

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

   4. lift-*.f64 N/A

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

   5. +-commutative N/A

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

   6. lift-*.f64 N/A

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

   7. lift-/.f64 N/A

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

   8. clear-num N/A

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

   9. un-div-inv N/A

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

   10. frac-2neg N/A

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

   11. associate-/r/ N/A

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

   12. 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 rewrites 96.5%

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

Alternative 9: 60.2% accurate, 6.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

   \[x + y \cdot \frac{z - t}{a - t} \]
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 60.6

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

5. Applied rewrites 60.6%

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

Alternative 10: 3.1% accurate, 8.7× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.0%

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

4. Applied rewrites 36.1%

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

6. Applied rewrites 24.3%

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

7. Taylor expanded in t around inf

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

   1. mul-1-neg N/A

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

   2. unsub-neg N/A

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

   3. lower--.f64 47.7

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

9. Applied rewrites 47.7%

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

10. Taylor expanded in x around 0

    \[\leadsto \color{blue}{-1 \cdot y} \]
11. Step-by-step derivation

    1. mul-1-neg N/A

       \[\leadsto \color{blue}{\mathsf{neg}\left(y\right)} \]

    2. lower-neg.f64 3.2

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

12. Applied rewrites 3.2%

    \[\leadsto \color{blue}{-y} \]
13. 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 2024220 
(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)))))