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

Percentage Accurate: 84.1% → 97.3%

Time: 7.5s

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: 84.1% 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: 97.3% accurate, 1.0× speedup

Math

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

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(x * Float64(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[(x * N[(N[(y - z), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 87.4%

   \[\frac{x \cdot \left(y - z\right)}{t - z} \]
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 98.4

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

4. Applied rewrites 98.4%

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

5. Final simplification 98.4%

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

Alternative 2: 68.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{z}, x, x\right)\\ \mathbf{if}\;z \leq -2.95 \cdot 10^{+35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 6 \cdot 10^{+52}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+71}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ t z) x x)))
   (if (<= z -2.95e+35)
     t_1
     (if (<= z 6e+52)
       (* (/ x (- t z)) y)
       (if (<= z 1.95e+71) (* (/ z t) (- x)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((t / z), x, x);
	double tmp;
	if (z <= -2.95e+35) {
		tmp = t_1;
	} else if (z <= 6e+52) {
		tmp = (x / (t - z)) * y;
	} else if (z <= 1.95e+71) {
		tmp = (z / t) * -x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(t / z), x, x)
	tmp = 0.0
	if (z <= -2.95e+35)
		tmp = t_1;
	elseif (z <= 6e+52)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	elseif (z <= 1.95e+71)
		tmp = Float64(Float64(z / t) * Float64(-x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[z, -2.95e+35], t$95$1, If[LessEqual[z, 6e+52], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[z, 1.95e+71], N[(N[(z / t), $MachinePrecision] * (-x)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.94999999999999993e35 or 1.9500000000000001e71 < z

   1. Initial program 78.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 83.3

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

   5. Applied rewrites 83.3%

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

   7. Applied rewrites 65.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 71.2%

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

   if -2.94999999999999993e35 < z < 6e52

   1. Initial program 94.4%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 74.0

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

   5. Applied rewrites 74.0%

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

   if 6e52 < z < 1.9500000000000001e71

   1. Initial program 87.8%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 87.5

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

   5. Applied rewrites 87.5%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 74.9%

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

Alternative 3: 60.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{z}, x, x\right)\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 7 \cdot 10^{-63}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+71}:\\ \;\;\;\;\frac{z}{t} \cdot \left(-x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ t z) x x)))
   (if (<= z -3.3e-35)
     t_1
     (if (<= z 7e-63)
       (* (/ y t) x)
       (if (<= z 1.95e+71) (* (/ z t) (- x)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((t / z), x, x);
	double tmp;
	if (z <= -3.3e-35) {
		tmp = t_1;
	} else if (z <= 7e-63) {
		tmp = (y / t) * x;
	} else if (z <= 1.95e+71) {
		tmp = (z / t) * -x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(t / z), x, x)
	tmp = 0.0
	if (z <= -3.3e-35)
		tmp = t_1;
	elseif (z <= 7e-63)
		tmp = Float64(Float64(y / t) * x);
	elseif (z <= 1.95e+71)
		tmp = Float64(Float64(z / t) * Float64(-x));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[z, -3.3e-35], t$95$1, If[LessEqual[z, 7e-63], N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.95e+71], N[(N[(z / t), $MachinePrecision] * (-x)), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3e-35 or 1.9500000000000001e71 < z

   1. Initial program 80.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 79.8

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

   5. Applied rewrites 79.8%

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

   7. Applied rewrites 64.1%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 68.1%

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

   if -3.3e-35 < z < 7.00000000000000006e-63

   1. Initial program 93.5%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 96.9

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

   4. Applied rewrites 96.9%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 73.0

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

   7. Applied rewrites 73.0%

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

   if 7.00000000000000006e-63 < z < 1.9500000000000001e71

   1. Initial program 94.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 61.3

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

   5. Applied rewrites 61.3%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 42.1%

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

Alternative 4: 61.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{z}, x, x\right)\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+48}:\\ \;\;\;\;\frac{-x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ t z) x x)))
   (if (<= z -3.3e-35)
     t_1
     (if (<= z 1.35e-19)
       (* (/ y t) x)
       (if (<= z 1.05e+48) (* (/ (- x) z) y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((t / z), x, x);
	double tmp;
	if (z <= -3.3e-35) {
		tmp = t_1;
	} else if (z <= 1.35e-19) {
		tmp = (y / t) * x;
	} else if (z <= 1.05e+48) {
		tmp = (-x / z) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(t / z), x, x)
	tmp = 0.0
	if (z <= -3.3e-35)
		tmp = t_1;
	elseif (z <= 1.35e-19)
		tmp = Float64(Float64(y / t) * x);
	elseif (z <= 1.05e+48)
		tmp = Float64(Float64(Float64(-x) / z) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[z, -3.3e-35], t$95$1, If[LessEqual[z, 1.35e-19], N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.05e+48], N[(N[((-x) / z), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3e-35 or 1.0499999999999999e48 < z

   1. Initial program 80.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 64.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 64.4%

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

   if -3.3e-35 < z < 1.35e-19

   1. Initial program 94.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 96.8

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

   4. Applied rewrites 96.8%

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

   5. Taylor expanded in z around 0

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

      1. lower-/.f64 69.3

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

   7. Applied rewrites 69.3%

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

   if 1.35e-19 < z < 1.0499999999999999e48

   1. Initial program 99.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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-lft-out N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      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. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 68.3

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

   5. Applied rewrites 68.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 62.5%

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

4. Final simplification 66.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{-35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, x, x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{-19}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+48}:\\ \;\;\;\;\frac{-x}{z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{z}, x, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 89.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{+157}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \mathbf{elif}\;z \leq 1.22 \cdot 10^{+126}:\\ \;\;\;\;\frac{x}{t - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.6e+157)
   (* (/ z (- z t)) x)
   (if (<= z 1.22e+126) (* (/ x (- t z)) (- y z)) (fma (- x) (/ y z) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.6e+157) {
		tmp = (z / (z - t)) * x;
	} else if (z <= 1.22e+126) {
		tmp = (x / (t - z)) * (y - z);
	} else {
		tmp = fma(-x, (y / z), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.6e+157)
		tmp = Float64(Float64(z / Float64(z - t)) * x);
	elseif (z <= 1.22e+126)
		tmp = Float64(Float64(x / Float64(t - z)) * Float64(y - z));
	else
		tmp = fma(Float64(-x), Float64(y / z), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.6e+157], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], If[LessEqual[z, 1.22e+126], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision], N[((-x) * N[(y / z), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.60000000000000011e157

   1. Initial program 73.9%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 99.1

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

   5. Applied rewrites 99.1%

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

   7. Applied rewrites 99.1%

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

   if -2.60000000000000011e157 < z < 1.21999999999999995e126

   1. Initial program 91.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. *-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 90.7

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

   4. Applied rewrites 90.7%

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

   if 1.21999999999999995e126 < z

   1. Initial program 82.7%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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-lft-out N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      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. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 82.5

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

   5. Applied rewrites 82.5%

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

   7. Applied rewrites 82.6%

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

Alternative 6: 75.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{t - z} \cdot x\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.25e-36)
   (fma (- x) (/ y z) x)
   (if (<= z 2.1e+47) (* (/ y (- t z)) x) (* (/ z (- z t)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.25e-36) {
		tmp = fma(-x, (y / z), x);
	} else if (z <= 2.1e+47) {
		tmp = (y / (t - z)) * x;
	} else {
		tmp = (z / (z - t)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.25e-36)
		tmp = fma(Float64(-x), Float64(y / z), x);
	elseif (z <= 2.1e+47)
		tmp = Float64(Float64(y / Float64(t - z)) * x);
	else
		tmp = Float64(Float64(z / Float64(z - t)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.25e-36], N[((-x) * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.1e+47], N[(N[(y / N[(t - z), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25000000000000001e-36

   1. Initial program 80.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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-lft-out N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      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. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 83.1%

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

   if -1.25000000000000001e-36 < z < 2.1e47

   1. Initial program 94.6%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 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 y around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 78.8

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

   7. Applied rewrites 78.8%

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

   if 2.1e47 < z

   1. Initial program 80.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 79.6%

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

Alternative 7: 74.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.25 \cdot 10^{-36}:\\ \;\;\;\;\mathsf{fma}\left(-x, \frac{y}{z}, x\right)\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{z - t} \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.25e-36)
   (fma (- x) (/ y z) x)
   (if (<= z 2.1e+47) (* (/ x (- t z)) y) (* (/ z (- z t)) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.25e-36) {
		tmp = fma(-x, (y / z), x);
	} else if (z <= 2.1e+47) {
		tmp = (x / (t - z)) * y;
	} else {
		tmp = (z / (z - t)) * x;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.25e-36)
		tmp = fma(Float64(-x), Float64(y / z), x);
	elseif (z <= 2.1e+47)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	else
		tmp = Float64(Float64(z / Float64(z - t)) * x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.25e-36], N[((-x) * N[(y / z), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 2.1e+47], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25000000000000001e-36

   1. Initial program 80.4%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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-lft-out N/A

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

      9. associate-/l* N/A

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

      10. *-commutative N/A

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

      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. lower-/.f64 N/A

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

      21. *-commutative N/A

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

      22. lower-*.f64 76.3

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

   5. Applied rewrites 76.3%

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

   7. Applied rewrites 83.1%

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

   if -1.25000000000000001e-36 < z < 2.1e47

   1. Initial program 94.6%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 76.6

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

   5. Applied rewrites 76.6%

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

   if 2.1e47 < z

   1. Initial program 80.0%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 79.6

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

   5. Applied rewrites 79.6%

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

   7. Applied rewrites 79.6%

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

Alternative 8: 75.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \frac{z}{z - t} \cdot x\\ \mathbf{if}\;z \leq -8:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.1 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{t - z} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (/ z (- z t)) x)))
   (if (<= z -8.0) t_1 (if (<= z 2.1e+47) (* (/ x (- t z)) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (z / (z - t)) * x;
	double tmp;
	if (z <= -8.0) {
		tmp = t_1;
	} else if (z <= 2.1e+47) {
		tmp = (x / (t - z)) * y;
	} 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 = (z / (z - t)) * x
    if (z <= (-8.0d0)) then
        tmp = t_1
    else if (z <= 2.1d+47) then
        tmp = (x / (t - z)) * y
    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 = (z / (z - t)) * x;
	double tmp;
	if (z <= -8.0) {
		tmp = t_1;
	} else if (z <= 2.1e+47) {
		tmp = (x / (t - z)) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (z / (z - t)) * x
	tmp = 0
	if z <= -8.0:
		tmp = t_1
	elif z <= 2.1e+47:
		tmp = (x / (t - z)) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(z / Float64(z - t)) * x)
	tmp = 0.0
	if (z <= -8.0)
		tmp = t_1;
	elseif (z <= 2.1e+47)
		tmp = Float64(Float64(x / Float64(t - z)) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (z / (z - t)) * x;
	tmp = 0.0;
	if (z <= -8.0)
		tmp = t_1;
	elseif (z <= 2.1e+47)
		tmp = (x / (t - z)) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(z / N[(z - t), $MachinePrecision]), $MachinePrecision] * x), $MachinePrecision]}, If[LessEqual[z, -8.0], t$95$1, If[LessEqual[z, 2.1e+47], N[(N[(x / N[(t - z), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -8 or 2.1e47 < z

   1. Initial program 79.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 82.2

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

   5. Applied rewrites 82.2%

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

   7. Applied rewrites 82.2%

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

   if -8 < z < 2.1e47

   1. Initial program 94.8%

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

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower--.f64 75.7

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

   5. Applied rewrites 75.7%

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

Alternative 9: 61.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{z}, x, x\right)\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{-35}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma (/ t z) x x)))
   (if (<= z -3.3e-35) t_1 (if (<= z 8.5e+47) (* (/ y t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma((t / z), x, x);
	double tmp;
	if (z <= -3.3e-35) {
		tmp = t_1;
	} else if (z <= 8.5e+47) {
		tmp = (y / t) * x;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(Float64(t / z), x, x)
	tmp = 0.0
	if (z <= -3.3e-35)
		tmp = t_1;
	elseif (z <= 8.5e+47)
		tmp = Float64(Float64(y / t) * x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t / z), $MachinePrecision] * x + x), $MachinePrecision]}, If[LessEqual[z, -3.3e-35], t$95$1, If[LessEqual[z, 8.5e+47], N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3e-35 or 8.5000000000000008e47 < z

   1. Initial program 80.1%

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

   3. Taylor expanded in y around 0

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

      1. associate-/l* N/A

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

      2. associate-*r* N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. mul-1-neg N/A

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

      8. lower-neg.f64 79.9

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

   5. Applied rewrites 79.9%

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

   7. Applied rewrites 64.6%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 64.4%

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

   if -3.3e-35 < z < 8.5000000000000008e47

   1. Initial program 94.6%

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

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

      1. lower-/.f64 65.3

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

   7. Applied rewrites 65.3%

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

Alternative 10: 61.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-36}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{y}{t} \cdot x\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.9e-36) (* 1.0 x) (if (<= z 8.5e+47) (* (/ y t) x) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e-36) {
		tmp = 1.0 * x;
	} else if (z <= 8.5e+47) {
		tmp = (y / t) * x;
	} else {
		tmp = 1.0 * x;
	}
	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 <= (-2.9d-36)) then
        tmp = 1.0d0 * x
    else if (z <= 8.5d+47) then
        tmp = (y / t) * x
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e-36) {
		tmp = 1.0 * x;
	} else if (z <= 8.5e+47) {
		tmp = (y / t) * x;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.9e-36:
		tmp = 1.0 * x
	elif z <= 8.5e+47:
		tmp = (y / t) * x
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.9e-36)
		tmp = Float64(1.0 * x);
	elseif (z <= 8.5e+47)
		tmp = Float64(Float64(y / t) * x);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.9e-36)
		tmp = 1.0 * x;
	elseif (z <= 8.5e+47)
		tmp = (y / t) * x;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.9e-36], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 8.5e+47], N[(N[(y / t), $MachinePrecision] * x), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.90000000000000013e-36 or 8.5000000000000008e47 < z

   1. Initial program 80.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 63.9%

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

   if -2.90000000000000013e-36 < z < 8.5000000000000008e47

   1. Initial program 94.6%

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

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

      1. lower-/.f64 65.3

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

   7. Applied rewrites 65.3%

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

Alternative 11: 60.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-36}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -2.9e-36) (* 1.0 x) (if (<= z 8.5e+47) (/ (* x y) t) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e-36) {
		tmp = 1.0 * x;
	} else if (z <= 8.5e+47) {
		tmp = (x * y) / t;
	} else {
		tmp = 1.0 * x;
	}
	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 <= (-2.9d-36)) then
        tmp = 1.0d0 * x
    else if (z <= 8.5d+47) then
        tmp = (x * y) / t
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -2.9e-36) {
		tmp = 1.0 * x;
	} else if (z <= 8.5e+47) {
		tmp = (x * y) / t;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -2.9e-36:
		tmp = 1.0 * x
	elif z <= 8.5e+47:
		tmp = (x * y) / t
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -2.9e-36)
		tmp = Float64(1.0 * x);
	elseif (z <= 8.5e+47)
		tmp = Float64(Float64(x * y) / t);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -2.9e-36)
		tmp = 1.0 * x;
	elseif (z <= 8.5e+47)
		tmp = (x * y) / t;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -2.9e-36], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 8.5e+47], N[(N[(x * y), $MachinePrecision] / t), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.90000000000000013e-36 or 8.5000000000000008e47 < z

   1. Initial program 80.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 63.9%

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

   if -2.90000000000000013e-36 < z < 8.5000000000000008e47

   1. Initial program 94.6%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. *-commutative N/A

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

      3. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

4. Final simplification 63.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.9 \cdot 10^{-36}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{x \cdot y}{t}\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 60.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.15 \cdot 10^{-36}:\\ \;\;\;\;1 \cdot x\\ \mathbf{elif}\;z \leq 8.5 \cdot 10^{+47}:\\ \;\;\;\;\frac{x}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;1 \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.15e-36) (* 1.0 x) (if (<= z 8.5e+47) (* (/ x t) y) (* 1.0 x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.15e-36) {
		tmp = 1.0 * x;
	} else if (z <= 8.5e+47) {
		tmp = (x / t) * y;
	} else {
		tmp = 1.0 * x;
	}
	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 <= (-1.15d-36)) then
        tmp = 1.0d0 * x
    else if (z <= 8.5d+47) then
        tmp = (x / t) * y
    else
        tmp = 1.0d0 * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.15e-36) {
		tmp = 1.0 * x;
	} else if (z <= 8.5e+47) {
		tmp = (x / t) * y;
	} else {
		tmp = 1.0 * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.15e-36:
		tmp = 1.0 * x
	elif z <= 8.5e+47:
		tmp = (x / t) * y
	else:
		tmp = 1.0 * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.15e-36)
		tmp = Float64(1.0 * x);
	elseif (z <= 8.5e+47)
		tmp = Float64(Float64(x / t) * y);
	else
		tmp = Float64(1.0 * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.15e-36)
		tmp = 1.0 * x;
	elseif (z <= 8.5e+47)
		tmp = (x / t) * y;
	else
		tmp = 1.0 * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.15e-36], N[(1.0 * x), $MachinePrecision], If[LessEqual[z, 8.5e+47], N[(N[(x / t), $MachinePrecision] * y), $MachinePrecision], N[(1.0 * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.14999999999999998e-36 or 8.5000000000000008e47 < z

   1. Initial program 80.1%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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 99.8

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

   4. Applied rewrites 99.8%

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

   5. Taylor expanded in z around inf

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

   7. Applied rewrites 63.9%

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

   if -1.14999999999999998e-36 < z < 8.5000000000000008e47

   1. Initial program 94.6%

      \[\frac{x \cdot \left(y - z\right)}{t - z} \]
   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. *-commutative N/A

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

      3. lower-*.f64 63.2

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

   5. Applied rewrites 63.2%

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

   7. Applied rewrites 61.7%

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

Alternative 13: 35.9% accurate, 3.8× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return 1.0 * 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 = 1.0d0 * x
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 87.4%

   \[\frac{x \cdot \left(y - z\right)}{t - z} \]
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 98.4

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

4. Applied rewrites 98.4%

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

5. Taylor expanded in z around inf

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

7. Applied rewrites 36.5%

   \[\leadsto \color{blue}{1} \cdot x \]
8. Add Preprocessing

Developer Target 1: 97.2% 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 2024243 
(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)))