Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 6.9s

Alternatives: 10

Speedup: 1.0×

• 34.6% 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

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{-27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.5 \cdot 10^{-164}:\\ \;\;\;\;\mathsf{fma}\left(-x, y, x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+21}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)))
   (if (<= z -1.7e-27)
     t_1
     (if (<= z -3.5e-164)
       (fma (- x) y x)
       (if (<= z 1.35e+21) (* (- t x) y) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - t) * z;
	double tmp;
	if (z <= -1.7e-27) {
		tmp = t_1;
	} else if (z <= -3.5e-164) {
		tmp = fma(-x, y, x);
	} else if (z <= 1.35e+21) {
		tmp = (t - x) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - t) * z)
	tmp = 0.0
	if (z <= -1.7e-27)
		tmp = t_1;
	elseif (z <= -3.5e-164)
		tmp = fma(Float64(-x), y, x);
	elseif (z <= 1.35e+21)
		tmp = Float64(Float64(t - x) * y);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.7e-27], t$95$1, If[LessEqual[z, -3.5e-164], N[((-x) * y + x), $MachinePrecision], If[LessEqual[z, 1.35e+21], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.69999999999999985e-27 or 1.35e21 < 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -1.69999999999999985e-27 < z < -3.5e-164

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 93.3

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

   5. Applied rewrites 93.3%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 70.9%

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

   if -3.5e-164 < z < 1.35e21

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 64.4

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

   5. Applied rewrites 64.4%

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

Alternative 3: 50.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -9.2 \cdot 10^{+157}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq -1.2 \cdot 10^{+35}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -9.2e+157)
   (* t y)
   (if (<= y -1.2e+35) (* (- x) y) (if (<= y 6e+56) (fma z x x) (* t y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -9.2e+157) {
		tmp = t * y;
	} else if (y <= -1.2e+35) {
		tmp = -x * y;
	} else if (y <= 6e+56) {
		tmp = fma(z, x, x);
	} else {
		tmp = t * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -9.2e+157)
		tmp = Float64(t * y);
	elseif (y <= -1.2e+35)
		tmp = Float64(Float64(-x) * y);
	elseif (y <= 6e+56)
		tmp = fma(z, x, x);
	else
		tmp = Float64(t * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -9.2e+157], N[(t * y), $MachinePrecision], If[LessEqual[y, -1.2e+35], N[((-x) * y), $MachinePrecision], If[LessEqual[y, 6e+56], N[(z * x + x), $MachinePrecision], N[(t * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -9.20000000000000015e157 or 6.00000000000000012e56 < 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 85.0

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

   5. Applied rewrites 85.0%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 56.2%

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

   if -9.20000000000000015e157 < y < -1.20000000000000007e35

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 75.9

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

   5. Applied rewrites 75.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto \left(-1 \cdot x\right) \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 61.3%

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

   if -1.20000000000000007e35 < y < 6.00000000000000012e56

   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. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 83.9

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

   7. Applied rewrites 83.9%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 53.3%

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

Alternative 4: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{-15}:\\ \;\;\;\;\mathsf{fma}\left(x - t, z, x\right)\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -6.2e-15)
   (fma (- x t) z x)
   (if (<= z 1.35e+21) (fma (- t x) y x) (* (- x t) z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -6.2e-15) {
		tmp = fma((x - t), z, x);
	} else if (z <= 1.35e+21) {
		tmp = fma((t - x), y, x);
	} else {
		tmp = (x - t) * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -6.2e-15)
		tmp = fma(Float64(x - t), z, x);
	elseif (z <= 1.35e+21)
		tmp = fma(Float64(t - x), y, x);
	else
		tmp = Float64(Float64(x - t) * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -6.2e-15], N[(N[(x - t), $MachinePrecision] * z + x), $MachinePrecision], If[LessEqual[z, 1.35e+21], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -6.1999999999999998e-15

   1. Initial program 100.0%

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

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   if -6.1999999999999998e-15 < z < 1.35e21

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 89.8

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

   5. Applied rewrites 89.8%

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

   if 1.35e21 < 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 83.9

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

   5. Applied rewrites 83.9%

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

Alternative 5: 85.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ \mathbf{if}\;z \leq -9.6 \cdot 10^{-7}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)))
   (if (<= z -9.6e-7) t_1 (if (<= z 1.35e+21) (fma (- t x) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - t) * z;
	double tmp;
	if (z <= -9.6e-7) {
		tmp = t_1;
	} else if (z <= 1.35e+21) {
		tmp = fma((t - x), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - t) * z)
	tmp = 0.0
	if (z <= -9.6e-7)
		tmp = t_1;
	elseif (z <= 1.35e+21)
		tmp = fma(Float64(t - x), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -9.6e-7], t$95$1, If[LessEqual[z, 1.35e+21], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -9.59999999999999914e-7 or 1.35e21 < 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 82.8

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

   5. Applied rewrites 82.8%

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

   if -9.59999999999999914e-7 < z < 1.35e21

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 88.8

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

   5. Applied rewrites 88.8%

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

Alternative 6: 69.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ \mathbf{if}\;z \leq -1.7 \cdot 10^{-27}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.35 \cdot 10^{+21}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)))
   (if (<= z -1.7e-27) t_1 (if (<= z 1.35e+21) (* (- t x) y) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - t) * z;
	double tmp;
	if (z <= -1.7e-27) {
		tmp = t_1;
	} else if (z <= 1.35e+21) {
		tmp = (t - x) * 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 = (x - t) * z
    if (z <= (-1.7d-27)) then
        tmp = t_1
    else if (z <= 1.35d+21) then
        tmp = (t - x) * 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 = (x - t) * z;
	double tmp;
	if (z <= -1.7e-27) {
		tmp = t_1;
	} else if (z <= 1.35e+21) {
		tmp = (t - x) * y;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x - t) * z
	tmp = 0
	if z <= -1.7e-27:
		tmp = t_1
	elif z <= 1.35e+21:
		tmp = (t - x) * y
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - t) * z)
	tmp = 0.0
	if (z <= -1.7e-27)
		tmp = t_1;
	elseif (z <= 1.35e+21)
		tmp = Float64(Float64(t - x) * y);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x - t) * z;
	tmp = 0.0;
	if (z <= -1.7e-27)
		tmp = t_1;
	elseif (z <= 1.35e+21)
		tmp = (t - x) * y;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -1.7e-27], t$95$1, If[LessEqual[z, 1.35e+21], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.69999999999999985e-27 or 1.35e21 < 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 80.9

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

   5. Applied rewrites 80.9%

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

   if -1.69999999999999985e-27 < z < 1.35e21

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 60.2

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

   5. Applied rewrites 60.2%

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

Alternative 7: 63.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1.2 \cdot 10^{+30}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{elif}\;x \leq 6 \cdot 10^{+22}:\\ \;\;\;\;t \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -1.2e+30) (fma z x x) (if (<= x 6e+22) (* t (- y z)) (fma z x x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -1.2e+30) {
		tmp = fma(z, x, x);
	} else if (x <= 6e+22) {
		tmp = t * (y - z);
	} else {
		tmp = fma(z, x, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -1.2e+30)
		tmp = fma(z, x, x);
	elseif (x <= 6e+22)
		tmp = Float64(t * Float64(y - z));
	else
		tmp = fma(z, x, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -1.2e+30], N[(z * x + x), $MachinePrecision], If[LessEqual[x, 6e+22], N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(z * x + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.2e30 or 6e22 < x

   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. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 95.7

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

   4. Applied rewrites 95.7%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 70.8

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

   7. Applied rewrites 70.8%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 61.3%

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

   if -1.2e30 < x < 6e22

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. lower-*.f64 N/A

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

      2. lower--.f64 73.7

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

   5. Applied rewrites 73.7%

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

Alternative 8: 49.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -5.8 \cdot 10^{+99}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq 6 \cdot 10^{+56}:\\ \;\;\;\;\mathsf{fma}\left(z, x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -5.8e+99) (* t y) (if (<= y 6e+56) (fma z x x) (* t y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -5.8e+99) {
		tmp = t * y;
	} else if (y <= 6e+56) {
		tmp = fma(z, x, x);
	} else {
		tmp = t * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -5.8e+99)
		tmp = Float64(t * y);
	elseif (y <= 6e+56)
		tmp = fma(z, x, x);
	else
		tmp = Float64(t * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -5.8e+99], N[(t * y), $MachinePrecision], If[LessEqual[y, 6e+56], N[(z * x + x), $MachinePrecision], N[(t * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -5.8000000000000004e99 or 6.00000000000000012e56 < 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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 84.4

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

   5. Applied rewrites 84.4%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 54.1%

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

   if -5.8000000000000004e99 < y < 6.00000000000000012e56

   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. *-commutative N/A

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

      5. lift--.f64 N/A

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

      6. sub-neg N/A

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

      7. distribute-lft-in N/A

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

      8. associate-+l+ N/A

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

      9. *-commutative N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-fma.f64 N/A

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

      12. lower-neg.f64 100.0

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

   4. Applied rewrites 100.0%

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

   5. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 80.4

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

   7. Applied rewrites 80.4%

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

   8. Taylor expanded in t around 0

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

   10. Applied rewrites 50.7%

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

Alternative 9: 39.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -66:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+84}:\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -66.0) (* z x) (if (<= z 1.05e+84) (* t y) (* z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -66.0) {
		tmp = z * x;
	} else if (z <= 1.05e+84) {
		tmp = t * y;
	} else {
		tmp = z * 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 <= (-66.0d0)) then
        tmp = z * x
    else if (z <= 1.05d+84) then
        tmp = t * y
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -66.0) {
		tmp = z * x;
	} else if (z <= 1.05e+84) {
		tmp = t * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -66.0:
		tmp = z * x
	elif z <= 1.05e+84:
		tmp = t * y
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -66.0)
		tmp = Float64(z * x);
	elseif (z <= 1.05e+84)
		tmp = Float64(t * y);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -66.0)
		tmp = z * x;
	elseif (z <= 1.05e+84)
		tmp = t * y;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -66.0], N[(z * x), $MachinePrecision], If[LessEqual[z, 1.05e+84], N[(t * y), $MachinePrecision], N[(z * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -66 or 1.05000000000000009e84 < 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. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 84.5

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

   5. Applied rewrites 84.5%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 47.9%

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

   if -66 < z < 1.05000000000000009e84

   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. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 34.9%

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

Alternative 10: 22.9% accurate, 2.5× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := N[(z * x), $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. *-commutative N/A

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

   2. associate-*r* N/A

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

   3. lower-*.f64 N/A

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

   4. mul-1-neg N/A

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

   5. sub-neg N/A

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

   6. +-commutative N/A

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

   7. distribute-neg-in N/A

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

   8. unsub-neg N/A

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

   9. remove-double-neg N/A

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

   10. lower--.f64 48.1

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

5. Applied rewrites 48.1%

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

6. Taylor expanded in t around 0

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

8. Applied rewrites 24.7%

   \[\leadsto z \cdot \color{blue}{x} \]
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 2024243 
(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))))