Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 9.1s

Alternatives: 10

Speedup: 1.2×

• 34.5% 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, 1.2× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

code[x_, y_, z_, t_] := N[(N[(t - x), $MachinePrecision] * N[(y - z), $MachinePrecision] + 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 x around 0

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

   1. +-commutative N/A

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

   2. distribute-lft-in N/A

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

   3. mul-1-neg N/A

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

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

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

   5. mul-1-neg N/A

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

   6. *-rgt-identity N/A

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

   7. associate-+l+ N/A

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

   8. +-commutative N/A

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

   9. +-commutative N/A

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

   10. mul-1-neg N/A

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

   11. unsub-neg N/A

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

   12. distribute-rgt-out-- N/A

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

   13. *-commutative N/A

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

   14. lower-fma.f64 N/A

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

   15. lower--.f64 N/A

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

   16. lower--.f64 100.0

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

5. Simplified 100.0%

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

Alternative 2: 68.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{-13}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.06 \cdot 10^{-150}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 92000000000:\\ \;\;\;\;z \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -3.2e-13)
     t_1
     (if (<= y 1.06e-150)
       (fma x z x)
       (if (<= y 92000000000.0) (* z (- x t)) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -3.2e-13) {
		tmp = t_1;
	} else if (y <= 1.06e-150) {
		tmp = fma(x, z, x);
	} else if (y <= 92000000000.0) {
		tmp = z * (x - t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	tmp = 0.0
	if (y <= -3.2e-13)
		tmp = t_1;
	elseif (y <= 1.06e-150)
		tmp = fma(x, z, x);
	elseif (y <= 92000000000.0)
		tmp = Float64(z * Float64(x - t));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -3.2e-13], t$95$1, If[LessEqual[y, 1.06e-150], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 92000000000.0], N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2e-13 or 9.2e10 < 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. *-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 78.7

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

   5. Simplified 78.7%

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

   if -3.2e-13 < y < 1.05999999999999996e-150

   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 73.4

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

   5. Simplified 73.4%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 73.4

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

   8. Simplified 73.4%

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

   if 1.05999999999999996e-150 < y < 9.2e10

   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 68.4

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

   5. Simplified 68.4%

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

Alternative 3: 49.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(-x\right)\\ \mathbf{if}\;y \leq -2.2 \cdot 10^{+263}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{+142}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 0.72:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- x))))
   (if (<= y -2.2e+263)
     (* t y)
     (if (<= y -1.55e+142) t_1 (if (<= y 0.72) (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 <= -2.2e+263) {
		tmp = t * y;
	} else if (y <= -1.55e+142) {
		tmp = t_1;
	} else if (y <= 0.72) {
		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 <= -2.2e+263)
		tmp = Float64(t * y);
	elseif (y <= -1.55e+142)
		tmp = t_1;
	elseif (y <= 0.72)
		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, -2.2e+263], N[(t * y), $MachinePrecision], If[LessEqual[y, -1.55e+142], t$95$1, If[LessEqual[y, 0.72], N[(x * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.2e263

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 68.2

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

   5. Simplified 68.2%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 67.5

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

   8. Simplified 67.5%

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

   if -2.2e263 < y < -1.55e142 or 0.71999999999999997 < 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. *-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 83.3

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

   5. Simplified 83.3%

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

   6. Taylor expanded in t around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 50.3

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

   8. Simplified 50.3%

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

   if -1.55e142 < y < 0.71999999999999997

   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 64.5

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

   5. Simplified 64.5%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 60.8

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

   8. Simplified 60.8%

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

4. Final simplification 57.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.2 \cdot 10^{+263}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq -1.55 \cdot 10^{+142}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \mathbf{elif}\;y \leq 0.72:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(-x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 46.8% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000105:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-100}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-48}:\\ \;\;\;\;-t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.000105)
   (fma x z x)
   (if (<= x -6e-100) (* t y) (if (<= x 1.26e-48) (- (* t z)) (fma x z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.000105) {
		tmp = fma(x, z, x);
	} else if (x <= -6e-100) {
		tmp = t * y;
	} else if (x <= 1.26e-48) {
		tmp = -(t * z);
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.000105)
		tmp = fma(x, z, x);
	elseif (x <= -6e-100)
		tmp = Float64(t * y);
	elseif (x <= 1.26e-48)
		tmp = Float64(-Float64(t * z));
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.000105], N[(x * z + x), $MachinePrecision], If[LessEqual[x, -6e-100], N[(t * y), $MachinePrecision], If[LessEqual[x, 1.26e-48], (-N[(t * z), $MachinePrecision]), N[(x * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -1.05e-4 or 1.2599999999999999e-48 < x

   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 84.7

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

   5. Simplified 84.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 60.9

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

   8. Simplified 60.9%

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

   if -1.05e-4 < x < -6.0000000000000001e-100

   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 59.6

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

   5. Simplified 59.6%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 46.0

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

   8. Simplified 46.0%

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

   if -6.0000000000000001e-100 < x < 1.2599999999999999e-48

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 77.9

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

   5. Simplified 77.9%

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

   6. Taylor expanded in y around 0

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

      1. associate-*r* N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. lower-neg.f64 47.7

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

   8. Simplified 47.7%

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

4. Final simplification 55.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000105:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;x \leq -6 \cdot 10^{-100}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;x \leq 1.26 \cdot 10^{-48}:\\ \;\;\;\;-t \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 83.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -6 \cdot 10^{+102}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 9 \cdot 10^{+22}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- x t))))
   (if (<= z -6e+102) t_1 (if (<= z 9e+22) (fma (- t x) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -6e+102) {
		tmp = t_1;
	} else if (z <= 9e+22) {
		tmp = fma((t - x), y, 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 <= -6e+102)
		tmp = t_1;
	elseif (z <= 9e+22)
		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[(z * N[(x - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -6e+102], t$95$1, If[LessEqual[z, 9e+22], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.9999999999999996e102 or 8.9999999999999996e22 < 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 83.6

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

   5. Simplified 83.6%

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

   if -5.9999999999999996e102 < z < 8.9999999999999996e22

   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 88.7

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

   5. Simplified 88.7%

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

Alternative 6: 76.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, z - y, x\right)\\ \mathbf{if}\;x \leq -2.7 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;x \leq 1.22 \cdot 10^{-17}:\\ \;\;\;\;t \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (fma x (- z y) x)))
   (if (<= x -2.7e-5) t_1 (if (<= x 1.22e-17) (* t (- y z)) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = fma(x, (z - y), x);
	double tmp;
	if (x <= -2.7e-5) {
		tmp = t_1;
	} else if (x <= 1.22e-17) {
		tmp = t * (y - z);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = fma(x, Float64(z - y), x)
	tmp = 0.0
	if (x <= -2.7e-5)
		tmp = t_1;
	elseif (x <= 1.22e-17)
		tmp = Float64(t * Float64(y - z));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(x * N[(z - y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[x, -2.7e-5], t$95$1, If[LessEqual[x, 1.22e-17], N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if x < -2.6999999999999999e-5 or 1.22e-17 < x

   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 87.6

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

   5. Simplified 87.6%

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

   if -2.6999999999999999e-5 < x < 1.22e-17

   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 74.0

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

   5. Simplified 74.0%

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

Alternative 7: 62.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.00038:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;t \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.00038)
   (fma x z x)
   (if (<= x 4.5e+45) (* t (- y z)) (fma x z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.00038) {
		tmp = fma(x, z, x);
	} else if (x <= 4.5e+45) {
		tmp = t * (y - z);
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.00038)
		tmp = fma(x, z, x);
	elseif (x <= 4.5e+45)
		tmp = Float64(t * Float64(y - z));
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.00038], N[(x * z + x), $MachinePrecision], If[LessEqual[x, 4.5e+45], N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision], N[(x * z + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -3.8000000000000002e-4 or 4.4999999999999998e45 < x

   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 90.7

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

   5. Simplified 90.7%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 65.4

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

   8. Simplified 65.4%

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

   if -3.8000000000000002e-4 < x < 4.4999999999999998e45

   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 70.6

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

   5. Simplified 70.6%

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

Alternative 8: 49.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+116}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -3.7e+116) (* t y) (if (<= y 1.5e-27) (fma x z x) (* t y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -3.7e+116) {
		tmp = t * y;
	} else if (y <= 1.5e-27) {
		tmp = fma(x, z, x);
	} else {
		tmp = t * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -3.7e+116)
		tmp = Float64(t * y);
	elseif (y <= 1.5e-27)
		tmp = fma(x, z, x);
	else
		tmp = Float64(t * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -3.7e+116], N[(t * y), $MachinePrecision], If[LessEqual[y, 1.5e-27], N[(x * z + x), $MachinePrecision], N[(t * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -3.7000000000000001e116 or 1.5000000000000001e-27 < y

   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 54.7

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

   5. Simplified 54.7%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 41.5

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

   8. Simplified 41.5%

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

   if -3.7000000000000001e116 < y < 1.5000000000000001e-27

   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 66.8

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

   5. Simplified 66.8%

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

   6. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. lower-fma.f64 62.8

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

   8. Simplified 62.8%

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

4. Final simplification 53.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.7 \cdot 10^{+116}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-27}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 35.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -0.000105:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -0.000105) (* x z) (if (<= x 4.5e+45) (* t y) (* x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -0.000105) {
		tmp = x * z;
	} else if (x <= 4.5e+45) {
		tmp = t * y;
	} 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 (x <= (-0.000105d0)) then
        tmp = x * z
    else if (x <= 4.5d+45) then
        tmp = t * y
    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 (x <= -0.000105) {
		tmp = x * z;
	} else if (x <= 4.5e+45) {
		tmp = t * y;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -0.000105:
		tmp = x * z
	elif x <= 4.5e+45:
		tmp = t * y
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -0.000105)
		tmp = Float64(x * z);
	elseif (x <= 4.5e+45)
		tmp = Float64(t * y);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -0.000105)
		tmp = x * z;
	elseif (x <= 4.5e+45)
		tmp = t * y;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -0.000105], N[(x * z), $MachinePrecision], If[LessEqual[x, 4.5e+45], N[(t * y), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if x < -1.05e-4 or 4.4999999999999998e45 < x

   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 43.3

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

   5. Simplified 43.3%

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

   6. Taylor expanded in x around inf

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

      1. lower-*.f64 40.1

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

   8. Simplified 40.1%

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

   if -1.05e-4 < x < 4.4999999999999998e45

   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 70.6

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

   5. Simplified 70.6%

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

   6. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 38.5

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

   8. Simplified 38.5%

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

4. Final simplification 39.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -0.000105:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;x \leq 4.5 \cdot 10^{+45}:\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 23.0% 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 45.8

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

5. Simplified 45.8%

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

6. Taylor expanded in x around inf

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

   1. lower-*.f64 25.5

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

8. Simplified 25.5%

   \[\leadsto \color{blue}{x \cdot z} \]
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 2024215 
(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))))