Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 9.0s

Alternatives: 13

Speedup: 1.0×

• 33.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - 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 = x + ((y - z) * (t - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - 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 = x + ((y - z) * (t - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

double code(double x, double y, double z, double t) {
	return x + ((y - z) * (t - 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 = x + ((y - z) * (t - x))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

   \[x + \left(y - z\right) \cdot \left(t - x\right) \]
2. Add Preprocessing
3. Add Preprocessing

Alternative 2: 68.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -35000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 2.4 \cdot 10^{-238}:\\ \;\;\;\;\mathsf{fma}\left(z, -t, x\right)\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- t x))))
   (if (<= y -35000000.0)
     t_1
     (if (<= y 2.4e-238)
       (fma z (- t) x)
       (if (<= y 6.4e+80) (fma x z x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double tmp;
	if (y <= -35000000.0) {
		tmp = t_1;
	} else if (y <= 2.4e-238) {
		tmp = fma(z, -t, x);
	} else if (y <= 6.4e+80) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	tmp = 0.0
	if (y <= -35000000.0)
		tmp = t_1;
	elseif (y <= 2.4e-238)
		tmp = fma(z, Float64(-t), x);
	elseif (y <= 6.4e+80)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -35000000.0], t$95$1, If[LessEqual[y, 2.4e-238], N[(z * (-t) + x), $MachinePrecision], If[LessEqual[y, 6.4e+80], N[(x * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.5e7 or 6.39999999999999979e80 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 81.6

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

   5. Applied rewrites 81.6%

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

   if -3.5e7 < y < 2.3999999999999998e-238

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 92.4

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

   7. Applied rewrites 92.4%

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

   8. Taylor expanded in x around 0

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

   10. Applied rewrites 74.4%

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

   if 2.3999999999999998e-238 < y < 6.39999999999999979e80

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 84.4

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

   7. Applied rewrites 84.4%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 66.0%

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

Alternative 3: 67.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -1.6 \cdot 10^{-209}:\\ \;\;\;\;y \cdot \left(t - x\right)\\ \mathbf{elif}\;z \leq 0.059:\\ \;\;\;\;\mathsf{fma}\left(y, -x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- x t))))
   (if (<= z -3.3e+68)
     t_1
     (if (<= z -1.6e-209)
       (* y (- t x))
       (if (<= z 0.059) (fma y (- x) x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -3.3e+68) {
		tmp = t_1;
	} else if (z <= -1.6e-209) {
		tmp = y * (t - x);
	} else if (z <= 0.059) {
		tmp = fma(y, -x, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(x - t))
	tmp = 0.0
	if (z <= -3.3e+68)
		tmp = t_1;
	elseif (z <= -1.6e-209)
		tmp = Float64(y * Float64(t - x));
	elseif (z <= 0.059)
		tmp = fma(y, Float64(-x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e+68], t$95$1, If[LessEqual[z, -1.6e-209], N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.059], N[(y * (-x) + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3e68 or 0.058999999999999997 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 80.7

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

   5. Applied rewrites 80.7%

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

   if -3.3e68 < z < -1.6000000000000001e-209

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 61.9

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

   5. Applied rewrites 61.9%

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

   if -1.6000000000000001e-209 < z < 0.058999999999999997

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 93.8

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

   5. Applied rewrites 93.8%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 71.2%

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

Alternative 4: 66.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -4.1 \cdot 10^{+46}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.05 \cdot 10^{-34}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- t x))))
   (if (<= y -4.1e+46)
     t_1
     (if (<= y -3.05e-34) (* (- y z) t) (if (<= y 6.4e+80) (fma x z x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double tmp;
	if (y <= -4.1e+46) {
		tmp = t_1;
	} else if (y <= -3.05e-34) {
		tmp = (y - z) * t;
	} else if (y <= 6.4e+80) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	tmp = 0.0
	if (y <= -4.1e+46)
		tmp = t_1;
	elseif (y <= -3.05e-34)
		tmp = Float64(Float64(y - z) * t);
	elseif (y <= 6.4e+80)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -4.1e+46], t$95$1, If[LessEqual[y, -3.05e-34], N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[y, 6.4e+80], N[(x * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -4.1e46 or 6.39999999999999979e80 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 84.0

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

   5. Applied rewrites 84.0%

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

   if -4.1e46 < y < -3.0499999999999999e-34

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 75.8

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

   5. Applied rewrites 75.8%

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

   if -3.0499999999999999e-34 < y < 6.39999999999999979e80

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 97.7

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

   4. Applied rewrites 97.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 88.8

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

   7. Applied rewrites 88.8%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 63.0%

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

4. Final simplification 72.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.1 \cdot 10^{+46}:\\ \;\;\;\;y \cdot \left(t - x\right)\\ \mathbf{elif}\;y \leq -3.05 \cdot 10^{-34}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{elif}\;y \leq 6.4 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(t - x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 50.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -1.6 \cdot 10^{+118}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4.7 \cdot 10^{+25}:\\ \;\;\;\;y \cdot t\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+82}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- x))))
   (if (<= y -1.6e+118)
     t_1
     (if (<= y -4.7e+25) (* y t) (if (<= y 8.2e+82) (fma x z x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * -x;
	double tmp;
	if (y <= -1.6e+118) {
		tmp = t_1;
	} else if (y <= -4.7e+25) {
		tmp = y * t;
	} else if (y <= 8.2e+82) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(-x))
	tmp = 0.0
	if (y <= -1.6e+118)
		tmp = t_1;
	elseif (y <= -4.7e+25)
		tmp = Float64(y * t);
	elseif (y <= 8.2e+82)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * (-x)), $MachinePrecision]}, If[LessEqual[y, -1.6e+118], t$95$1, If[LessEqual[y, -4.7e+25], N[(y * t), $MachinePrecision], If[LessEqual[y, 8.2e+82], N[(x * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.60000000000000008e118 or 8.1999999999999999e82 < y

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around inf

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

      1. mul-1-neg N/A

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

      2. sub-neg N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 88.3

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

   7. Applied rewrites 88.3%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 56.2%

       \[\leadsto y \cdot \left(-x\right) \]

   if -1.60000000000000008e118 < y < -4.6999999999999998e25

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 66.2

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

   5. Applied rewrites 66.2%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 45.6%

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

   if -4.6999999999999998e25 < y < 8.1999999999999999e82

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 87.7

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 59.8%

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

Alternative 6: 47.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(-t\right)\\ \mathbf{if}\;t \leq -1.8 \cdot 10^{+106}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 960000000:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;t \leq 10^{+215}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- t))))
   (if (<= t -1.8e+106)
     t_1
     (if (<= t 960000000.0) (fma x z x) (if (<= t 1e+215) t_1 (* y t))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * -t;
	double tmp;
	if (t <= -1.8e+106) {
		tmp = t_1;
	} else if (t <= 960000000.0) {
		tmp = fma(x, z, x);
	} else if (t <= 1e+215) {
		tmp = t_1;
	} else {
		tmp = y * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(-t))
	tmp = 0.0
	if (t <= -1.8e+106)
		tmp = t_1;
	elseif (t <= 960000000.0)
		tmp = fma(x, z, x);
	elseif (t <= 1e+215)
		tmp = t_1;
	else
		tmp = Float64(y * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * (-t)), $MachinePrecision]}, If[LessEqual[t, -1.8e+106], t$95$1, If[LessEqual[t, 960000000.0], N[(x * z + x), $MachinePrecision], If[LessEqual[t, 1e+215], t$95$1, N[(y * t), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -1.8e106 or 9.6e8 < t < 9.99999999999999907e214

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 54.9

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

   5. Applied rewrites 54.9%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 51.8%

      \[\leadsto z \cdot \left(-t\right) \]

   if -1.8e106 < t < 9.6e8

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 62.7

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

   7. Applied rewrites 62.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 56.2%

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

   if 9.99999999999999907e214 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 83.7

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

   5. Applied rewrites 83.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 77.8%

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

Alternative 7: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.3 \cdot 10^{+68}:\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{elif}\;z \leq 0.0265:\\ \;\;\;\;\mathsf{fma}\left(y, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x - t, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -3.3e+68)
   (* z (- x t))
   (if (<= z 0.0265) (fma y (- t x) x) (fma z (- x t) x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -3.3e+68) {
		tmp = z * (x - t);
	} else if (z <= 0.0265) {
		tmp = fma(y, (t - x), x);
	} else {
		tmp = fma(z, (x - t), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -3.3e+68)
		tmp = Float64(z * Float64(x - t));
	elseif (z <= 0.0265)
		tmp = fma(y, Float64(t - x), x);
	else
		tmp = fma(z, Float64(x - t), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -3.3e+68], N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 0.0265], N[(y * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(z * N[(x - t), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.3e68

   1. Initial program 99.9%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 82.0

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

   5. Applied rewrites 82.0%

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

   if -3.3e68 < z < 0.0264999999999999993

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 88.8

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

   5. Applied rewrites 88.8%

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

   if 0.0264999999999999993 < z

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. remove-double-neg N/A

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

      12. lower--.f64 82.0

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

   5. Applied rewrites 82.0%

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

Alternative 8: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -3.3 \cdot 10^{+68}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 5.8 \cdot 10^{+50}:\\ \;\;\;\;\mathsf{fma}\left(y, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- x t))))
   (if (<= z -3.3e+68) t_1 (if (<= z 5.8e+50) (fma y (- t x) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -3.3e+68) {
		tmp = t_1;
	} else if (z <= 5.8e+50) {
		tmp = fma(y, (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(x - t))
	tmp = 0.0
	if (z <= -3.3e+68)
		tmp = t_1;
	elseif (z <= 5.8e+50)
		tmp = fma(y, Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -3.3e+68], t$95$1, If[LessEqual[z, 5.8e+50], N[(y * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.3e68 or 5.8e50 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 84.8

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

   5. Applied rewrites 84.8%

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

   if -3.3e68 < z < 5.8e50

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 85.1

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

   5. Applied rewrites 85.1%

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

Alternative 9: 76.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot t\\ \mathbf{if}\;t \leq -3.6 \cdot 10^{+82}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(x, z - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) t)))
   (if (<= t -3.6e+82) t_1 (if (<= t 7e-41) (fma x (- z y) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * t;
	double tmp;
	if (t <= -3.6e+82) {
		tmp = t_1;
	} else if (t <= 7e-41) {
		tmp = fma(x, (z - y), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * t)
	tmp = 0.0
	if (t <= -3.6e+82)
		tmp = t_1;
	elseif (t <= 7e-41)
		tmp = fma(x, Float64(z - y), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -3.6e+82], t$95$1, If[LessEqual[t, 7e-41], N[(x * N[(z - y), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -3.60000000000000014e82 or 6.9999999999999999e-41 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 78.5

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

   5. Applied rewrites 78.5%

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

   if -3.60000000000000014e82 < t < 6.9999999999999999e-41

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

      3. *-rgt-identity N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 83.1

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

   5. Applied rewrites 83.1%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3.6 \cdot 10^{+82}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{elif}\;t \leq 7 \cdot 10^{-41}:\\ \;\;\;\;\mathsf{fma}\left(x, z - y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 62.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot t\\ \mathbf{if}\;t \leq -2.65 \cdot 10^{+79}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) t)))
   (if (<= t -2.65e+79) t_1 (if (<= t 2.25e-56) (fma x z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * t;
	double tmp;
	if (t <= -2.65e+79) {
		tmp = t_1;
	} else if (t <= 2.25e-56) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * t)
	tmp = 0.0
	if (t <= -2.65e+79)
		tmp = t_1;
	elseif (t <= 2.25e-56)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[t, -2.65e+79], t$95$1, If[LessEqual[t, 2.25e-56], N[(x * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.64999999999999989e79 or 2.25e-56 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 76.3

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

   5. Applied rewrites 76.3%

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

   if -2.64999999999999989e79 < t < 2.25e-56

   1. Initial program 100.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 66.1

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

   7. Applied rewrites 66.1%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 60.1%

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

4. Final simplification 67.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.65 \cdot 10^{+79}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-56}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 50.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.7 \cdot 10^{+25}:\\ \;\;\;\;y \cdot t\\ \mathbf{elif}\;y \leq 9.2 \cdot 10^{+80}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -4.7e+25) (* y t) (if (<= y 9.2e+80) (fma x z x) (* y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -4.7e+25) {
		tmp = y * t;
	} else if (y <= 9.2e+80) {
		tmp = fma(x, z, x);
	} else {
		tmp = y * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -4.7e+25)
		tmp = Float64(y * t);
	elseif (y <= 9.2e+80)
		tmp = fma(x, z, x);
	else
		tmp = Float64(y * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -4.7e+25], N[(y * t), $MachinePrecision], If[LessEqual[y, 9.2e+80], N[(x * z + x), $MachinePrecision], N[(y * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -4.6999999999999998e25 or 9.20000000000000016e80 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 43.2%

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

   if -4.6999999999999998e25 < y < 9.20000000000000016e80

   1. Initial program 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 97.9

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

   4. Applied rewrites 97.9%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. +-commutative N/A

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

      3. associate-*r* N/A

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

      4. distribute-rgt-in N/A

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

      5. lower-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. lower--.f64 87.7

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

   7. Applied rewrites 87.7%

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

   8. Taylor expanded in x around inf

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

   10. Applied rewrites 59.8%

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

Alternative 12: 38.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.4 \cdot 10^{+19}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+55}:\\ \;\;\;\;y \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -7.4e+19) (* x z) (if (<= z 2.7e+55) (* y t) (* x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.4e+19) {
		tmp = x * z;
	} else if (z <= 2.7e+55) {
		tmp = y * t;
	} else {
		tmp = 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 (z <= (-7.4d+19)) then
        tmp = x * z
    else if (z <= 2.7d+55) then
        tmp = y * t
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -7.4e+19) {
		tmp = x * z;
	} else if (z <= 2.7e+55) {
		tmp = y * t;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -7.4e+19:
		tmp = x * z
	elif z <= 2.7e+55:
		tmp = y * t
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -7.4e+19)
		tmp = Float64(x * z);
	elseif (z <= 2.7e+55)
		tmp = Float64(y * t);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -7.4e+19)
		tmp = x * z;
	elseif (z <= 2.7e+55)
		tmp = y * t;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -7.4e+19], N[(x * z), $MachinePrecision], If[LessEqual[z, 2.7e+55], N[(y * t), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -7.4e19 or 2.69999999999999977e55 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

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

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

      3. mul-1-neg N/A

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

      4. lower-*.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 81.6

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

   5. Applied rewrites 81.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 49.9%

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

   if -7.4e19 < z < 2.69999999999999977e55

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 N/A

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

      3. lower--.f64 86.6

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

   5. Applied rewrites 86.6%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 33.4%

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

Alternative 13: 22.2% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Taylor expanded in z around inf

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

   1. mul-1-neg N/A

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

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

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

   3. mul-1-neg N/A

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

   4. lower-*.f64 N/A

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

   5. mul-1-neg N/A

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

   6. sub-neg N/A

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

   7. +-commutative N/A

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

   8. distribute-neg-in N/A

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

   9. unsub-neg N/A

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

   10. remove-double-neg N/A

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

   11. lower--.f64 43.1

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

5. Applied rewrites 43.1%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 23.6%

   \[\leadsto x \cdot \color{blue}{z} \]
9. Add Preprocessing

Developer Target 1: 96.5% accurate, 0.6× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024233 
(FPCore (x y z t)
  :name "Data.Metrics.Snapshot:quantile from metrics-0.3.0.2"
  :precision binary64

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

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