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

Percentage Accurate: 98.0% → 98.0%

Time: 10.2s

Alternatives: 11

Speedup: 1.1×

Specification

Math

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

FPCore

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

C

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

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) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.0% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. frac-2neg N/A

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

   4. frac-2neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lower-fma.f64 99.5

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

4. Applied egg-rr 99.5%

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

Alternative 2: 65.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \frac{t}{a}\\ t_2 := \frac{z - t}{z - a} \cdot y\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+276}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+226}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* y (/ t a))) (t_2 (* (/ (- z t) (- z a)) y)))
   (if (<= t_2 -5e+276) t_1 (if (<= t_2 1e+226) (+ y x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (t / a);
	double t_2 = ((z - t) / (z - a)) * y;
	double tmp;
	if (t_2 <= -5e+276) {
		tmp = t_1;
	} else if (t_2 <= 1e+226) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: t_1
    real(8) :: t_2
    real(8) :: tmp
    t_1 = y * (t / a)
    t_2 = ((z - t) / (z - a)) * y
    if (t_2 <= (-5d+276)) then
        tmp = t_1
    else if (t_2 <= 1d+226) then
        tmp = y + x
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = y * (t / a);
	double t_2 = ((z - t) / (z - a)) * y;
	double tmp;
	if (t_2 <= -5e+276) {
		tmp = t_1;
	} else if (t_2 <= 1e+226) {
		tmp = y + x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = y * (t / a)
	t_2 = ((z - t) / (z - a)) * y
	tmp = 0
	if t_2 <= -5e+276:
		tmp = t_1
	elif t_2 <= 1e+226:
		tmp = y + x
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(y * Float64(t / a))
	t_2 = Float64(Float64(Float64(z - t) / Float64(z - a)) * y)
	tmp = 0.0
	if (t_2 <= -5e+276)
		tmp = t_1;
	elseif (t_2 <= 1e+226)
		tmp = Float64(y + x);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = y * (t / a);
	t_2 = ((z - t) / (z - a)) * y;
	tmp = 0.0;
	if (t_2 <= -5e+276)
		tmp = t_1;
	elseif (t_2 <= 1e+226)
		tmp = y + x;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(t / a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+276], t$95$1, If[LessEqual[t$95$2, 1e+226], N[(y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < -5.00000000000000001e276 or 9.99999999999999961e225 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a)))

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 60.1

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

   5. Simplified 60.1%

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

   6. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 55.7

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

   8. Simplified 55.7%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 66.0

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

   10. Applied egg-rr 66.0%

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

   if -5.00000000000000001e276 < (*.f64 y (/.f64 (-.f64 z t) (-.f64 z a))) < 9.99999999999999961e225

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 67.7

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

   5. Simplified 67.7%

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

4. Final simplification 67.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{z - t}{z - a} \cdot y \leq -5 \cdot 10^{+276}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \mathbf{elif}\;\frac{z - t}{z - a} \cdot y \leq 10^{+226}:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 87.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 100:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(z - t\right) \cdot \frac{y}{z - a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-13)
     (fma y (/ (- t z) a) x)
     (if (<= t_1 100.0) (fma y (/ z (- z a)) x) (* (- z t) (/ y (- z a)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 1e-13) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 100.0) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = (z - t) * (y / (z - a));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 1e-13)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 100.0)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = Float64(Float64(z - t) * Float64(y / Float64(z - a)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-13], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 100.0], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(z - t), $MachinePrecision] * N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-13

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 90.1

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

   5. Simplified 90.1%

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

   if 1e-13 < (/.f64 (-.f64 z t) (-.f64 z a)) < 100

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.5

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

   5. Simplified 99.5%

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

   if 100 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.6%

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

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. frac-2neg N/A

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

      4. frac-2neg N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. +-commutative N/A

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

      8. lift-*.f64 N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 97.6

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

   4. Applied egg-rr 97.6%

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

   5. Taylor expanded in y around inf

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

      1. div-sub N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 67.2

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

   7. Simplified 67.2%

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

Alternative 4: 87.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - z}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 100:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-13)
     (fma y (/ (- t z) a) x)
     (if (<= t_1 100.0) (fma y (/ z (- z a)) x) (* t (/ y (- a z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 1e-13) {
		tmp = fma(y, ((t - z) / a), x);
	} else if (t_1 <= 100.0) {
		tmp = fma(y, (z / (z - a)), x);
	} else {
		tmp = t * (y / (a - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 1e-13)
		tmp = fma(y, Float64(Float64(t - z) / a), x);
	elseif (t_1 <= 100.0)
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	else
		tmp = Float64(t * Float64(y / Float64(a - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-13], N[(y * N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 100.0], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-13

   1. Initial program 99.8%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. mul-1-neg N/A

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

      6. lower-fma.f64 N/A

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

      7. associate-*r/ N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. sub-neg N/A

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

      11. +-commutative N/A

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

      12. distribute-neg-in N/A

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

      13. unsub-neg N/A

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

      14. remove-double-neg N/A

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

      15. lower--.f64 90.1

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

   5. Simplified 90.1%

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

   if 1e-13 < (/.f64 (-.f64 z t) (-.f64 z a)) < 100

   1. Initial program 100.0%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 99.5

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

   5. Simplified 99.5%

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

   if 100 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. neg-mul-1 N/A

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

      11. lower-+.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 58.8

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

   5. Simplified 58.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 66.7

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 66.7

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

   7. Applied egg-rr 66.7%

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

Alternative 5: 82.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 100:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-13)
     (fma y (/ t a) x)
     (if (<= t_1 100.0) (+ y x) (* t (/ y (- a z)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 1e-13) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 100.0) {
		tmp = y + x;
	} else {
		tmp = t * (y / (a - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 1e-13)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 100.0)
		tmp = Float64(y + x);
	else
		tmp = Float64(t * Float64(y / Float64(a - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-13], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 100.0], N[(y + x), $MachinePrecision], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-13

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 77.5

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

   5. Simplified 77.5%

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

   if 1e-13 < (/.f64 (-.f64 z t) (-.f64 z a)) < 100

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 98.9

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

   5. Simplified 98.9%

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

   if 100 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.6%

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

   3. Taylor expanded in t around inf

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. lower-/.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. remove-double-neg N/A

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

      10. neg-mul-1 N/A

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

      11. lower-+.f64 N/A

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

      12. neg-mul-1 N/A

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

      13. lower-neg.f64 58.8

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

   5. Simplified 58.8%

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

      1. lift-neg.f64 N/A

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

      2. lift-+.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 66.7

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

      7. lift-+.f64 N/A

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

      8. lift-neg.f64 N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 66.7

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

   7. Applied egg-rr 66.7%

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

Alternative 6: 81.3% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ \mathbf{if}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))))
   (if (<= t_1 1e-13)
     (fma y (/ t a) x)
     (if (<= t_1 500000.0) (+ y x) (fma t (/ y a) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double tmp;
	if (t_1 <= 1e-13) {
		tmp = fma(y, (t / a), x);
	} else if (t_1 <= 500000.0) {
		tmp = y + x;
	} else {
		tmp = fma(t, (y / a), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	tmp = 0.0
	if (t_1 <= 1e-13)
		tmp = fma(y, Float64(t / a), x);
	elseif (t_1 <= 500000.0)
		tmp = Float64(y + x);
	else
		tmp = fma(t, Float64(y / a), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-13], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 500000.0], N[(y + x), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-13

   1. Initial program 99.8%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 77.5

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

   5. Simplified 77.5%

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

   if 1e-13 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.9

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

   5. Simplified 97.9%

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

   if 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 97.5%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 59.6

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

   5. Simplified 59.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 65.7

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

   7. Applied egg-rr 65.7%

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

Alternative 7: 81.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(t, \frac{y}{a}, x\right)\\ \mathbf{if}\;t\_1 \leq 10^{-13}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 500000:\\ \;\;\;\;y + x\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma t (/ y a) x)))
   (if (<= t_1 1e-13) t_2 (if (<= t_1 500000.0) (+ y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma(t, (y / a), x);
	double tmp;
	if (t_1 <= 1e-13) {
		tmp = t_2;
	} else if (t_1 <= 500000.0) {
		tmp = y + x;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(t, Float64(y / a), x)
	tmp = 0.0
	if (t_1 <= 1e-13)
		tmp = t_2;
	elseif (t_1 <= 500000.0)
		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[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(t * N[(y / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t$95$1, 1e-13], t$95$2, If[LessEqual[t$95$1, 500000.0], N[(y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 1e-13 or 5e5 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 99.2%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 69.9

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

   5. Simplified 69.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. *-commutative N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 72.6

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

   7. Applied egg-rr 72.6%

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

   if 1e-13 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5e5

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. lower-+.f64 97.9

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

   5. Simplified 97.9%

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

Alternative 8: 82.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{-30}:\\ \;\;\;\;\mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{elif}\;z \leq 1.45 \cdot 10^{-109}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{z}{z - a}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.4e-30)
   (fma y (- 1.0 (/ t z)) x)
   (if (<= z 1.45e-109) (fma y (/ t a) x) (fma y (/ z (- z a)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.4e-30) {
		tmp = fma(y, (1.0 - (t / z)), x);
	} else if (z <= 1.45e-109) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = fma(y, (z / (z - a)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.4e-30)
		tmp = fma(y, Float64(1.0 - Float64(t / z)), x);
	elseif (z <= 1.45e-109)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = fma(y, Float64(z / Float64(z - a)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.4e-30], N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.45e-109], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], N[(y * N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.39999999999999985e-30

   1. Initial program 99.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 84.1

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

   5. Simplified 84.1%

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

   if -2.39999999999999985e-30 < z < 1.45e-109

   1. Initial program 98.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 90.9

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

   5. Simplified 90.9%

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

   if 1.45e-109 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 81.5

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

   5. Simplified 81.5%

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

Alternative 9: 82.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y, 1 - \frac{t}{z}, x\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{-30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{-82}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma y (- 1.0 (/ t z)) x)))
   (if (<= z -2.4e-30) t_1 (if (<= z 1.22e-82) (fma y (/ t a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(y, (1.0 - (t / z)), x);
	double tmp;
	if (z <= -2.4e-30) {
		tmp = t_1;
	} else if (z <= 1.22e-82) {
		tmp = fma(y, (t / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(y, Float64(1.0 - Float64(t / z)), x)
	tmp = 0.0
	if (z <= -2.4e-30)
		tmp = t_1;
	elseif (z <= 1.22e-82)
		tmp = fma(y, Float64(t / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(y * N[(1.0 - N[(t / z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -2.4e-30], t$95$1, If[LessEqual[z, 1.22e-82], N[(y * N[(t / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.39999999999999985e-30 or 1.22000000000000001e-82 < z

   1. Initial program 99.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. div-sub N/A

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

      4. sub-neg N/A

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

      5. *-inverses N/A

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

      6. mul-1-neg N/A

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

      7. lower-fma.f64 N/A

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

      8. mul-1-neg N/A

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

      9. unsub-neg N/A

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

      10. lower--.f64 N/A

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

      11. lower-/.f64 80.1

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

   5. Simplified 80.1%

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

   if -2.39999999999999985e-30 < z < 1.22000000000000001e-82

   1. Initial program 99.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 89.8

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

   5. Simplified 89.8%

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

Alternative 10: 95.8% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 99.5%

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

   1. lift--.f64 N/A

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

   2. lift--.f64 N/A

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

   3. frac-2neg N/A

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

   4. frac-2neg N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. +-commutative N/A

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

   8. lift-*.f64 N/A

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

   9. *-commutative N/A

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

   10. lift-/.f64 N/A

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

   11. div-inv N/A

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

   12. associate-*l* N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. associate-*l/ N/A

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

   16. *-lft-identity N/A

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

   17. lower-/.f64 97.0

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

4. Applied egg-rr 97.0%

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

Alternative 11: 60.9% 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 99.5%

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

3. Taylor expanded in z around inf

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

   1. +-commutative N/A

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

   2. lower-+.f64 59.2

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

5. Simplified 59.2%

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

Developer Target 1: 98.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

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, A"
  :precision binary64

  :alt
  (! :herbie-platform default (+ x (/ y (/ (- z a) (- z t)))))

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