Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3

Percentage Accurate: 85.1% → 97.2%

Time: 11.2s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 97.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 82.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 96.3

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

4. Applied rewrites 96.3%

   \[\leadsto \color{blue}{\frac{y - z}{t - z} \cdot x} \]
5. Add Preprocessing

Alternative 2: 34.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot x}{t - z} \leq 5 \cdot 10^{+236}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ (* (- y z) x) (- t z)) 5e+236) (* x 1.0) (* z (/ x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((((y - z) * x) / (t - z)) <= 5e+236) {
		tmp = x * 1.0;
	} else {
		tmp = z * (x / z);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if ((((y - z) * x) / (t - z)) <= 5d+236) then
        tmp = x * 1.0d0
    else
        tmp = z * (x / z)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((((y - z) * x) / (t - z)) <= 5e+236) {
		tmp = x * 1.0;
	} else {
		tmp = z * (x / z);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (((y - z) * x) / (t - z)) <= 5e+236:
		tmp = x * 1.0
	else:
		tmp = z * (x / z)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(Float64(Float64(y - z) * x) / Float64(t - z)) <= 5e+236)
		tmp = Float64(x * 1.0);
	else
		tmp = Float64(z * Float64(x / z));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((((y - z) * x) / (t - z)) <= 5e+236)
		tmp = x * 1.0;
	else
		tmp = z * (x / z);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision], 5e+236], N[(x * 1.0), $MachinePrecision], N[(z * N[(x / z), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z)) < 4.9999999999999997e236

   1. Initial program 88.4%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.0

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

   4. Applied rewrites 97.0%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 33.6%

      \[\leadsto \color{blue}{1} \cdot x \]

   if 4.9999999999999997e236 < (/.f64 (*.f64 x (-.f64 y z)) (-.f64 t z))

   1. Initial program 39.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 90.8

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

   4. Applied rewrites 90.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-*r* N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. sub-neg N/A

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

      18. lift--.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 96.7%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 48.9

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

   9. Applied rewrites 48.9%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 50.4%

       \[\leadsto z \cdot \frac{x}{\color{blue}{z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 35.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{\left(y - z\right) \cdot x}{t - z} \leq 5 \cdot 10^{+236}:\\ \;\;\;\;x \cdot 1\\ \mathbf{else}:\\ \;\;\;\;z \cdot \frac{x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 66.1% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ t_2 := z \cdot \frac{x}{z - t}\\ \mathbf{if}\;z \leq -1.35 \cdot 10^{+191}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-62}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+184}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (/ t z) x)) (t_2 (* z (/ x (- z t)))))
   (if (<= z -1.35e+191)
     t_1
     (if (<= z -3.3e-62)
       t_2
       (if (<= z 1.15e-71) (* x (/ y t)) (if (<= z 1.6e+184) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, (t / z), x);
	double t_2 = z * (x / (z - t));
	double tmp;
	if (z <= -1.35e+191) {
		tmp = t_1;
	} else if (z <= -3.3e-62) {
		tmp = t_2;
	} else if (z <= 1.15e-71) {
		tmp = x * (y / t);
	} else if (z <= 1.6e+184) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(x, Float64(t / z), x)
	t_2 = Float64(z * Float64(x / Float64(z - t)))
	tmp = 0.0
	if (z <= -1.35e+191)
		tmp = t_1;
	elseif (z <= -3.3e-62)
		tmp = t_2;
	elseif (z <= 1.15e-71)
		tmp = Float64(x * Float64(y / t));
	elseif (z <= 1.6e+184)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(t / z), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(z * N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.35e+191], t$95$1, If[LessEqual[z, -3.3e-62], t$95$2, If[LessEqual[z, 1.15e-71], N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.6e+184], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.34999999999999998e191 or 1.59999999999999991e184 < z

   1. Initial program 55.0%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-*r* N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. sub-neg N/A

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

      18. lift--.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 70.4%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 62.5

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

   9. Applied rewrites 62.5%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 82.9%

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

   if -1.34999999999999998e191 < z < -3.30000000000000004e-62 or 1.1499999999999999e-71 < z < 1.59999999999999991e184

   1. Initial program 87.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.5

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

   4. Applied rewrites 99.5%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-*r* N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. sub-neg N/A

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

      18. lift--.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 90.1%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 60.1

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

   9. Applied rewrites 60.1%

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

   if -3.30000000000000004e-62 < z < 1.1499999999999999e-71

   1. Initial program 92.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 91.0

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

   4. Applied rewrites 91.0%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 67.7

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

   7. Applied rewrites 67.7%

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

4. Final simplification 67.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+191}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ \mathbf{elif}\;z \leq -3.3 \cdot 10^{-62}:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-71}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{elif}\;z \leq 1.6 \cdot 10^{+184}:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 75.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{y}{-z}, x\right)\\ \mathbf{if}\;z \leq -1.18 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (/ y (- z)) x)))
   (if (<= z -1.18e+23)
     t_1
     (if (<= z -1.1e-89)
       (/ (* (- y z) x) t)
       (if (<= z 8e+21) (* x (/ y (- t z))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, (y / -z), x);
	double tmp;
	if (z <= -1.18e+23) {
		tmp = t_1;
	} else if (z <= -1.1e-89) {
		tmp = ((y - z) * x) / t;
	} else if (z <= 8e+21) {
		tmp = x * (y / (t - z));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(x, Float64(y / Float64(-z)), x)
	tmp = 0.0
	if (z <= -1.18e+23)
		tmp = t_1;
	elseif (z <= -1.1e-89)
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	elseif (z <= 8e+21)
		tmp = Float64(x * Float64(y / Float64(t - z)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y / (-z)), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[z, -1.18e+23], t$95$1, If[LessEqual[z, -1.1e-89], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[z, 8e+21], N[(x * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.18e23 or 8e21 < z

   1. Initial program 68.9%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-/.f64 76.1

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

   5. Applied rewrites 76.1%

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

   7. Applied rewrites 70.6%

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

   9. Applied rewrites 79.5%

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

   if -1.18e23 < z < -1.10000000000000006e-89

   1. Initial program 99.7%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if -1.10000000000000006e-89 < z < 8e21

   1. Initial program 92.3%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 77.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.18 \cdot 10^{+23}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{-z}, x\right)\\ \mathbf{elif}\;z \leq -1.1 \cdot 10^{-89}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;z \leq 8 \cdot 10^{+21}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{-z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 70.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{t - z}\\ \mathbf{if}\;y \leq -2.4 \cdot 10^{+18}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-146}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+33}:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ y (- t z)))))
   (if (<= y -2.4e+18)
     t_1
     (if (<= y -2.4e-146)
       (/ (* (- y z) x) t)
       (if (<= y 3.1e+33) (* z (/ x (- z t))) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y / (t - z));
	double tmp;
	if (y <= -2.4e+18) {
		tmp = t_1;
	} else if (y <= -2.4e-146) {
		tmp = ((y - z) * x) / t;
	} else if (y <= 3.1e+33) {
		tmp = z * (x / (z - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y / (t - z))
    if (y <= (-2.4d+18)) then
        tmp = t_1
    else if (y <= (-2.4d-146)) then
        tmp = ((y - z) * x) / t
    else if (y <= 3.1d+33) then
        tmp = z * (x / (z - 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 t_1 = x * (y / (t - z));
	double tmp;
	if (y <= -2.4e+18) {
		tmp = t_1;
	} else if (y <= -2.4e-146) {
		tmp = ((y - z) * x) / t;
	} else if (y <= 3.1e+33) {
		tmp = z * (x / (z - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (y / (t - z))
	tmp = 0
	if y <= -2.4e+18:
		tmp = t_1
	elif y <= -2.4e-146:
		tmp = ((y - z) * x) / t
	elif y <= 3.1e+33:
		tmp = z * (x / (z - t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y / Float64(t - z)))
	tmp = 0.0
	if (y <= -2.4e+18)
		tmp = t_1;
	elseif (y <= -2.4e-146)
		tmp = Float64(Float64(Float64(y - z) * x) / t);
	elseif (y <= 3.1e+33)
		tmp = Float64(z * Float64(x / Float64(z - t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y / (t - z));
	tmp = 0.0;
	if (y <= -2.4e+18)
		tmp = t_1;
	elseif (y <= -2.4e-146)
		tmp = ((y - z) * x) / t;
	elseif (y <= 3.1e+33)
		tmp = z * (x / (z - t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.4e+18], t$95$1, If[LessEqual[y, -2.4e-146], N[(N[(N[(y - z), $MachinePrecision] * x), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[y, 3.1e+33], N[(z * N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.4e18 or 3.1e33 < y

   1. Initial program 81.6%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 75.8

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

   5. Applied rewrites 75.8%

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

   if -2.4e18 < y < -2.4000000000000002e-146

   1. Initial program 94.5%

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

   3. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 65.8

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

   5. Applied rewrites 65.8%

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

   if -2.4000000000000002e-146 < y < 3.1e33

   1. Initial program 79.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.6

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

   4. Applied rewrites 97.6%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-*r* N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. sub-neg N/A

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

      18. lift--.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 88.0%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 81.7

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

   9. Applied rewrites 81.7%

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

4. Final simplification 76.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.4 \cdot 10^{+18}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{elif}\;y \leq -2.4 \cdot 10^{-146}:\\ \;\;\;\;\frac{\left(y - z\right) \cdot x}{t}\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+33}:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 89.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+213}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+182}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{-z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.4e+213)
   (* x (/ z (- z t)))
   (if (<= z 3.1e+182) (* (- y z) (/ x (- t z))) (fma x (/ y (- z)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.4e+213) {
		tmp = x * (z / (z - t));
	} else if (z <= 3.1e+182) {
		tmp = (y - z) * (x / (t - z));
	} else {
		tmp = fma(x, (y / -z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.4e+213)
		tmp = Float64(x * Float64(z / Float64(z - t)));
	elseif (z <= 3.1e+182)
		tmp = Float64(Float64(y - z) * Float64(x / Float64(t - z)));
	else
		tmp = fma(x, Float64(y / Float64(-z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.4e+213], N[(x * N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 3.1e+182], N[(N[(y - z), $MachinePrecision] * N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(x * N[(y / (-z)), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4e213

   1. Initial program 54.6%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-in N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 97.2

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

   7. Applied rewrites 97.2%

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

   if -2.4e213 < z < 3.09999999999999996e182

   1. Initial program 88.8%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 N/A

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

      7. lower-/.f64 90.9

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

   4. Applied rewrites 90.9%

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

   if 3.09999999999999996e182 < z

   1. Initial program 59.0%

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

   3. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. neg-sub0 N/A

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

      3. associate-/l* N/A

         \[\leadsto 0 - \color{blue}{x \cdot \frac{y - z}{z}} \]

      4. div-sub N/A

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

      5. sub-neg N/A

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

      6. *-inverses N/A

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

      7. metadata-eval N/A

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

      8. distribute-rgt-in N/A

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

      9. *-commutative N/A

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

      10. associate-/l* N/A

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

      11. mul-1-neg N/A

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

      12. unsub-neg N/A

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

      13. associate-+l- N/A

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

      14. neg-sub0 N/A

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

      15. mul-1-neg N/A

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

      16. +-commutative N/A

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

      17. mul-1-neg N/A

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

      18. unsub-neg N/A

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

      19. lower--.f64 N/A

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

      20. *-commutative N/A

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

      21. associate-/l* N/A

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

      22. lower-*.f64 N/A

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

      23. lower-/.f64 85.3

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

   5. Applied rewrites 85.3%

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

   7. Applied rewrites 85.6%

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

   9. Applied rewrites 88.8%

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

4. Final simplification 91.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.4 \cdot 10^{+213}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{elif}\;z \leq 3.1 \cdot 10^{+182}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{x}{t - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{y}{-z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 76.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{t - z}\\ \mathbf{if}\;y \leq -2 \cdot 10^{+59}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ y (- t z)))))
   (if (<= y -2e+59) t_1 (if (<= y 9.6e+35) (* x (/ z (- z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y / (t - z));
	double tmp;
	if (y <= -2e+59) {
		tmp = t_1;
	} else if (y <= 9.6e+35) {
		tmp = x * (z / (z - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y / (t - z))
    if (y <= (-2d+59)) then
        tmp = t_1
    else if (y <= 9.6d+35) then
        tmp = x * (z / (z - 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 t_1 = x * (y / (t - z));
	double tmp;
	if (y <= -2e+59) {
		tmp = t_1;
	} else if (y <= 9.6e+35) {
		tmp = x * (z / (z - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (y / (t - z))
	tmp = 0
	if y <= -2e+59:
		tmp = t_1
	elif y <= 9.6e+35:
		tmp = x * (z / (z - t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y / Float64(t - z)))
	tmp = 0.0
	if (y <= -2e+59)
		tmp = t_1;
	elseif (y <= 9.6e+35)
		tmp = Float64(x * Float64(z / Float64(z - t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y / (t - z));
	tmp = 0.0;
	if (y <= -2e+59)
		tmp = t_1;
	elseif (y <= 9.6e+35)
		tmp = x * (z / (z - t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2e+59], t$95$1, If[LessEqual[y, 9.6e+35], N[(x * N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.99999999999999994e59 or 9.60000000000000058e35 < y

   1. Initial program 81.9%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 79.1

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

   5. Applied rewrites 79.1%

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

   if -1.99999999999999994e59 < y < 9.60000000000000058e35

   1. Initial program 82.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.6

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

   4. Applied rewrites 97.6%

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

   5. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. distribute-neg-frac2 N/A

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

      3. mul-1-neg N/A

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

      4. sub-neg N/A

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

      5. mul-1-neg N/A

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

      6. distribute-lft-in N/A

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

      7. mul-1-neg N/A

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

      8. neg-mul-1 N/A

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

      9. mul-1-neg N/A

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

      10. remove-double-neg N/A

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

      11. +-commutative N/A

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

      12. sub-neg N/A

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

      13. lower-/.f64 N/A

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

      14. lower--.f64 81.0

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

   7. Applied rewrites 81.0%

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

4. Final simplification 80.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2 \cdot 10^{+59}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \mathbf{elif}\;y \leq 9.6 \cdot 10^{+35}:\\ \;\;\;\;x \cdot \frac{z}{z - t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{y}{t - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot \frac{y}{t - z}\\ \mathbf{if}\;y \leq -3 \cdot 10^{-104}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{+33}:\\ \;\;\;\;z \cdot \frac{x}{z - t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* x (/ y (- t z)))))
   (if (<= y -3e-104) t_1 (if (<= y 3.1e+33) (* z (/ x (- z t))) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = x * (y / (t - z));
	double tmp;
	if (y <= -3e-104) {
		tmp = t_1;
	} else if (y <= 3.1e+33) {
		tmp = z * (x / (z - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = x * (y / (t - z))
    if (y <= (-3d-104)) then
        tmp = t_1
    else if (y <= 3.1d+33) then
        tmp = z * (x / (z - 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 t_1 = x * (y / (t - z));
	double tmp;
	if (y <= -3e-104) {
		tmp = t_1;
	} else if (y <= 3.1e+33) {
		tmp = z * (x / (z - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = x * (y / (t - z))
	tmp = 0
	if y <= -3e-104:
		tmp = t_1
	elif y <= 3.1e+33:
		tmp = z * (x / (z - t))
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(x * Float64(y / Float64(t - z)))
	tmp = 0.0
	if (y <= -3e-104)
		tmp = t_1;
	elseif (y <= 3.1e+33)
		tmp = Float64(z * Float64(x / Float64(z - t)));
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = x * (y / (t - z));
	tmp = 0.0;
	if (y <= -3e-104)
		tmp = t_1;
	elseif (y <= 3.1e+33)
		tmp = z * (x / (z - t));
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -3e-104], t$95$1, If[LessEqual[y, 3.1e+33], N[(z * N[(x / N[(z - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.0000000000000002e-104 or 3.1e33 < y

   1. Initial program 84.4%

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

   3. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 71.2

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

   5. Applied rewrites 71.2%

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

   if -3.0000000000000002e-104 < y < 3.1e33

   1. Initial program 79.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 97.8

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

   4. Applied rewrites 97.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-*r* N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. sub-neg N/A

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

      18. lift--.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 88.1%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 79.6

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

   9. Applied rewrites 79.6%

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

Alternative 9: 61.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.5e+47)
   (* x 1.0)
   (if (<= z 1.8e+22) (* x (/ y t)) (fma x (/ t z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+47) {
		tmp = x * 1.0;
	} else if (z <= 1.8e+22) {
		tmp = x * (y / t);
	} else {
		tmp = fma(x, (t / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.5e+47)
		tmp = Float64(x * 1.0);
	elseif (z <= 1.8e+22)
		tmp = Float64(x * Float64(y / t));
	else
		tmp = fma(x, Float64(t / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.5e+47], N[(x * 1.0), $MachinePrecision], If[LessEqual[z, 1.8e+22], N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision], N[(x * N[(t / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.50000000000000015e47

   1. Initial program 63.2%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 68.4%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -3.50000000000000015e47 < z < 1.8e22

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 57.0

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

   7. Applied rewrites 57.0%

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

   if 1.8e22 < z

   1. Initial program 72.5%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. *-commutative N/A

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

      5. lift-*.f64 N/A

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

      6. frac-2neg N/A

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

      7. *-lft-identity N/A

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

      8. associate-*l/ N/A

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

      9. lift-*.f64 N/A

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

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

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

      11. associate-*r* N/A

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

      12. lift--.f64 N/A

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

      13. sub-neg N/A

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

      14. +-commutative N/A

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

      15. distribute-neg-in N/A

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

      16. remove-double-neg N/A

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

      17. sub-neg N/A

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

      18. lift--.f64 N/A

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

      19. lower-*.f64 N/A

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

   6. Applied rewrites 82.1%

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

   7. Taylor expanded in y around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 64.5

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

   9. Applied rewrites 64.5%

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

   10. Taylor expanded in z around inf

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

   12. Applied rewrites 57.8%

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

4. Final simplification 59.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{t}{z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 61.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.5e+47) (* x 1.0) (if (<= z 1.8e+22) (* x (/ y t)) (* x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+47) {
		tmp = x * 1.0;
	} else if (z <= 1.8e+22) {
		tmp = x * (y / t);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.5d+47)) then
        tmp = x * 1.0d0
    else if (z <= 1.8d+22) then
        tmp = x * (y / t)
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+47) {
		tmp = x * 1.0;
	} else if (z <= 1.8e+22) {
		tmp = x * (y / t);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.5e+47:
		tmp = x * 1.0
	elif z <= 1.8e+22:
		tmp = x * (y / t)
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.5e+47)
		tmp = Float64(x * 1.0);
	elseif (z <= 1.8e+22)
		tmp = Float64(x * Float64(y / t));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.5e+47)
		tmp = x * 1.0;
	elseif (z <= 1.8e+22)
		tmp = x * (y / t);
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.5e+47], N[(x * 1.0), $MachinePrecision], If[LessEqual[z, 1.8e+22], N[(x * N[(y / t), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.50000000000000015e47 or 1.8e22 < z

   1. Initial program 68.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 62.6%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -3.50000000000000015e47 < z < 1.8e22

   1. Initial program 93.9%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 93.5

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

   4. Applied rewrites 93.5%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 57.0

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

   7. Applied rewrites 57.0%

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

4. Final simplification 59.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+22}:\\ \;\;\;\;x \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 59.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+44}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -8.5e+44) (* x 1.0) (if (<= z 8.4e+21) (/ (* y x) t) (* x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.5e+44) {
		tmp = x * 1.0;
	} else if (z <= 8.4e+21) {
		tmp = (y * x) / t;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-8.5d+44)) then
        tmp = x * 1.0d0
    else if (z <= 8.4d+21) then
        tmp = (y * x) / t
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -8.5e+44) {
		tmp = x * 1.0;
	} else if (z <= 8.4e+21) {
		tmp = (y * x) / t;
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -8.5e+44:
		tmp = x * 1.0
	elif z <= 8.4e+21:
		tmp = (y * x) / t
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -8.5e+44)
		tmp = Float64(x * 1.0);
	elseif (z <= 8.4e+21)
		tmp = Float64(Float64(y * x) / t);
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -8.5e+44)
		tmp = x * 1.0;
	elseif (z <= 8.4e+21)
		tmp = (y * x) / t;
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -8.5e+44], N[(x * 1.0), $MachinePrecision], If[LessEqual[z, 8.4e+21], N[(N[(y * x), $MachinePrecision] / t), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -8.5e44 or 8.4e21 < z

   1. Initial program 68.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 62.6%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -8.5e44 < z < 8.4e21

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

4. Final simplification 58.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -8.5 \cdot 10^{+44}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;z \leq 8.4 \cdot 10^{+21}:\\ \;\;\;\;\frac{y \cdot x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 59.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+22}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.5e+47) (* x 1.0) (if (<= z 1.8e+22) (* y (/ x t)) (* x 1.0))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+47) {
		tmp = x * 1.0;
	} else if (z <= 1.8e+22) {
		tmp = y * (x / t);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-3.5d+47)) then
        tmp = x * 1.0d0
    else if (z <= 1.8d+22) then
        tmp = y * (x / t)
    else
        tmp = x * 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.5e+47) {
		tmp = x * 1.0;
	} else if (z <= 1.8e+22) {
		tmp = y * (x / t);
	} else {
		tmp = x * 1.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -3.5e+47:
		tmp = x * 1.0
	elif z <= 1.8e+22:
		tmp = y * (x / t)
	else:
		tmp = x * 1.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.5e+47)
		tmp = Float64(x * 1.0);
	elseif (z <= 1.8e+22)
		tmp = Float64(y * Float64(x / t));
	else
		tmp = Float64(x * 1.0);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -3.5e+47)
		tmp = x * 1.0;
	elseif (z <= 1.8e+22)
		tmp = y * (x / t);
	else
		tmp = x * 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.5e+47], N[(x * 1.0), $MachinePrecision], If[LessEqual[z, 1.8e+22], N[(y * N[(x / t), $MachinePrecision]), $MachinePrecision], N[(x * 1.0), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.50000000000000015e47 or 1.8e22 < z

   1. Initial program 68.1%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 99.9

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

   4. Applied rewrites 99.9%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 62.6%

      \[\leadsto \color{blue}{1} \cdot x \]

   if -3.50000000000000015e47 < z < 1.8e22

   1. Initial program 93.9%

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

   3. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 55.8

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

   5. Applied rewrites 55.8%

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

   7. Applied rewrites 54.2%

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

4. Final simplification 57.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+47}:\\ \;\;\;\;x \cdot 1\\ \mathbf{elif}\;z \leq 1.8 \cdot 10^{+22}:\\ \;\;\;\;y \cdot \frac{x}{t}\\ \mathbf{else}:\\ \;\;\;\;x \cdot 1\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 33.7% accurate, 3.8× speedup

Math

\[\begin{array}{l} \\ x \cdot 1 \end{array} \]

FPCore

(FPCore (x y z t) :precision binary64 (* x 1.0))

C

double code(double x, double y, double z, double t) {
	return x * 1.0;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = x * 1.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return x * 1.0;
}

Python

def code(x, y, z, t):
	return x * 1.0

Julia

function code(x, y, z, t)
	return Float64(x * 1.0)
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = x * 1.0;
end

Wolfram

code[x_, y_, z_, t_] := N[(x * 1.0), $MachinePrecision]

Derivation

1. Initial program 82.5%

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

   1. lift-/.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. associate-/l* N/A

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

   4. *-commutative N/A

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

   5. lower-*.f64 N/A

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

   6. lower-/.f64 96.3

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

4. Applied rewrites 96.3%

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

5. Taylor expanded in z around inf

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

7. Applied rewrites 33.9%

   \[\leadsto \color{blue}{1} \cdot x \]

8. Final simplification 33.9%

   \[\leadsto x \cdot 1 \]
9. Add Preprocessing

Developer Target 1: 97.1% accurate, 0.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024220 
(FPCore (x y z t)
  :name "Graphics.Rendering.Chart.Plot.AreaSpots:renderAreaSpots4D from Chart-1.5.3"
  :precision binary64

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

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