Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 10.5s

Alternatives: 11

Speedup: 1.0×

• 34.0% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

Alternative 2: 73.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(y - z\right) \cdot t\\ t_2 := \mathsf{fma}\left(x, z - y, x\right)\\ \mathbf{if}\;t \leq -2.6 \cdot 10^{+162}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.75 \cdot 10^{+62}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t \leq -100000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{-33}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- y z) t)) (t_2 (fma x (- z y) x)))
   (if (<= t -2.6e+162)
     t_1
     (if (<= t -1.75e+62)
       t_2
       (if (<= t -100000000000.0) t_1 (if (<= t 5.2e-33) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (y - z) * t;
	double t_2 = fma(x, (z - y), x);
	double tmp;
	if (t <= -2.6e+162) {
		tmp = t_1;
	} else if (t <= -1.75e+62) {
		tmp = t_2;
	} else if (t <= -100000000000.0) {
		tmp = t_1;
	} else if (t <= 5.2e-33) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(y - z) * t)
	t_2 = fma(x, Float64(z - y), x)
	tmp = 0.0
	if (t <= -2.6e+162)
		tmp = t_1;
	elseif (t <= -1.75e+62)
		tmp = t_2;
	elseif (t <= -100000000000.0)
		tmp = t_1;
	elseif (t <= 5.2e-33)
		tmp = t_2;
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]}, Block[{t$95$2 = N[(x * N[(z - y), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[t, -2.6e+162], t$95$1, If[LessEqual[t, -1.75e+62], t$95$2, If[LessEqual[t, -100000000000.0], t$95$1, If[LessEqual[t, 5.2e-33], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 2 regimes

2. if t < -2.6e162 or -1.74999999999999992e62 < t < -1e11 or 5.19999999999999988e-33 < t

   1. Initial program 100.0%

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

   3. Taylor expanded in x around 0

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

      1. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, y - z, 0\right)} \]

      3. --lowering--.f64 86.0

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{y - z}, 0\right) \]

   5. Simplified 86.0%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, y - z, 0\right)} \]
   6. Step-by-step derivation

      1. +-rgt-identity N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

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

      4. --lowering--.f64 86.0

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

   7. Applied egg-rr 86.0%

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

   if -2.6e162 < t < -1.74999999999999992e62 or -1e11 < t < 5.19999999999999988e-33

   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. accelerator-lowering-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. --lowering--.f64 78.7

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

   5. Simplified 78.7%

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

Alternative 3: 71.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.75 \cdot 10^{-182}:\\ \;\;\;\;y \cdot \left(t - x\right)\\ \mathbf{elif}\;z \leq 560000000:\\ \;\;\;\;\mathsf{fma}\left(y, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- x t))))
   (if (<= z -5.5e+62)
     t_1
     (if (<= z -2.75e-182)
       (* y (- t x))
       (if (<= z 560000000.0) (fma y t x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -5.5e+62) {
		tmp = t_1;
	} else if (z <= -2.75e-182) {
		tmp = y * (t - x);
	} else if (z <= 560000000.0) {
		tmp = fma(y, t, 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 <= -5.5e+62)
		tmp = t_1;
	elseif (z <= -2.75e-182)
		tmp = Float64(y * Float64(t - x));
	elseif (z <= 560000000.0)
		tmp = fma(y, t, 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, -5.5e+62], t$95$1, If[LessEqual[z, -2.75e-182], N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 560000000.0], N[(y * t + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -5.4999999999999997e62 or 5.6e8 < 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. *-lowering-*.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. --lowering--.f64 82.2

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

   5. Simplified 82.2%

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

   if -5.4999999999999997e62 < z < -2.74999999999999996e-182

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 100.0

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

   4. Applied egg-rr 100.0%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 74.4

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

   7. Simplified 74.4%

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

   if -2.74999999999999996e-182 < z < 5.6e8

   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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 89.9

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

   5. Simplified 89.9%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 73.2%

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

Alternative 4: 84.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -5000000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.75 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(z, x - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- t x))))
   (if (<= y -5000000000000.0)
     t_1
     (if (<= y 3.75e+34) (fma z (- x t) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double tmp;
	if (y <= -5000000000000.0) {
		tmp = t_1;
	} else if (y <= 3.75e+34) {
		tmp = fma(z, (x - t), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	tmp = 0.0
	if (y <= -5000000000000.0)
		tmp = t_1;
	elseif (y <= 3.75e+34)
		tmp = fma(z, Float64(x - t), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -5000000000000.0], t$95$1, If[LessEqual[y, 3.75e+34], N[(z * N[(x - t), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -5e12 or 3.74999999999999988e34 < y

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 93.9

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

   4. Applied egg-rr 93.9%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 81.1

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

   7. Simplified 81.1%

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

   if -5e12 < y < 3.74999999999999988e34

   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. mul-1-neg N/A

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

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

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

      4. mul-1-neg N/A

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

      5. accelerator-lowering-fma.f64 N/A

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

      6. mul-1-neg N/A

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

      7. sub-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. remove-double-neg N/A

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

      12. --lowering--.f64 89.9

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

   5. Simplified 89.9%

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

Alternative 5: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \left(x - t\right)\\ \mathbf{if}\;z \leq -5.5 \cdot 10^{+62}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 3000000000:\\ \;\;\;\;\mathsf{fma}\left(y, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (- x t))))
   (if (<= z -5.5e+62) t_1 (if (<= z 3000000000.0) (fma y (- t x) x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * (x - t);
	double tmp;
	if (z <= -5.5e+62) {
		tmp = t_1;
	} else if (z <= 3000000000.0) {
		tmp = fma(y, (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(x - t))
	tmp = 0.0
	if (z <= -5.5e+62)
		tmp = t_1;
	elseif (z <= 3000000000.0)
		tmp = fma(y, Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(x - t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -5.5e+62], t$95$1, If[LessEqual[z, 3000000000.0], N[(y * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999997e62 or 3e9 < 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. *-lowering-*.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. --lowering--.f64 82.2

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

   5. Simplified 82.2%

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

   if -5.4999999999999997e62 < z < 3e9

   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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 86.9

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

   5. Simplified 86.9%

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

Alternative 6: 67.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := y \cdot \left(t - x\right)\\ \mathbf{if}\;y \leq -82:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* y (- t x))))
   (if (<= y -82.0) t_1 (if (<= y 3.7e+34) (fma x z x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = y * (t - x);
	double tmp;
	if (y <= -82.0) {
		tmp = t_1;
	} else if (y <= 3.7e+34) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(y * Float64(t - x))
	tmp = 0.0
	if (y <= -82.0)
		tmp = t_1;
	elseif (y <= 3.7e+34)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(y * N[(t - x), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -82.0], t$95$1, If[LessEqual[y, 3.7e+34], N[(x * z + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -82 or 3.70000000000000009e34 < y

   1. Initial program 100.0%

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

      1. *-commutative N/A

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

      2. sub-neg N/A

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

      3. +-commutative N/A

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

      4. distribute-lft-in N/A

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

      5. *-commutative N/A

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

      6. accelerator-lowering-fma.f64 N/A

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

      7. --lowering--.f64 N/A

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

      8. neg-sub0 N/A

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

      9. --lowering--.f64 N/A

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

      10. *-lowering-*.f64 N/A

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

      11. --lowering--.f64 93.9

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

   4. Applied egg-rr 93.9%

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

   5. Taylor expanded in y around inf

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

      1. *-lowering-*.f64 N/A

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

      2. --lowering--.f64 80.4

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

   7. Simplified 80.4%

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

   if -82 < y < 3.70000000000000009e34

   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. accelerator-lowering-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. --lowering--.f64 59.1

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

   5. Simplified 59.1%

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

   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. accelerator-lowering-fma.f64 57.7

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

   8. Simplified 57.7%

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

Alternative 7: 54.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+62}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 540000000000:\\ \;\;\;\;\mathsf{fma}\left(y, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.5e+62) (* x z) (if (<= z 540000000000.0) (fma y t x) (* x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.5e+62) {
		tmp = x * z;
	} else if (z <= 540000000000.0) {
		tmp = fma(y, t, x);
	} else {
		tmp = x * z;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.5e+62)
		tmp = Float64(x * z);
	elseif (z <= 540000000000.0)
		tmp = fma(y, t, x);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.5e+62], N[(x * z), $MachinePrecision], If[LessEqual[z, 540000000000.0], N[(y * t + x), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999997e62 or 5.4e11 < 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. *-lowering-*.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. --lowering--.f64 82.2

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

   5. Simplified 82.2%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 46.6%

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

   if -5.4999999999999997e62 < z < 5.4e11

   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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 86.9

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

   5. Simplified 86.9%

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

   6. Taylor expanded in t around inf

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

   8. Simplified 63.6%

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

4. Final simplification 55.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+62}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 540000000000:\\ \;\;\;\;\mathsf{fma}\left(y, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 50.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -42:\\ \;\;\;\;y \cdot t\\ \mathbf{elif}\;y \leq 3.7 \cdot 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -42.0) (* y t) (if (<= y 3.7e+34) (fma x z x) (* y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -42.0) {
		tmp = y * t;
	} else if (y <= 3.7e+34) {
		tmp = fma(x, z, x);
	} else {
		tmp = y * t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -42.0)
		tmp = Float64(y * t);
	elseif (y <= 3.7e+34)
		tmp = fma(x, z, x);
	else
		tmp = Float64(y * t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -42.0], N[(y * t), $MachinePrecision], If[LessEqual[y, 3.7e+34], N[(x * z + x), $MachinePrecision], N[(y * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -42 or 3.70000000000000009e34 < 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. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, y - z, 0\right)} \]

      3. --lowering--.f64 57.5

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{y - z}, 0\right) \]

   5. Simplified 57.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, y - z, 0\right)} \]

   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. *-lowering-*.f64 46.5

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

   8. Simplified 46.5%

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

   if -42 < y < 3.70000000000000009e34

   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. accelerator-lowering-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. --lowering--.f64 59.1

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

   5. Simplified 59.1%

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

   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. accelerator-lowering-fma.f64 57.7

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

   8. Simplified 57.7%

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

Alternative 9: 39.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+62}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 12000000000:\\ \;\;\;\;y \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -5.5e+62) (* x z) (if (<= z 12000000000.0) (* y t) (* x z))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.5e+62) {
		tmp = x * z;
	} else if (z <= 12000000000.0) {
		tmp = y * t;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-5.5d+62)) then
        tmp = x * z
    else if (z <= 12000000000.0d0) then
        tmp = y * t
    else
        tmp = x * z
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -5.5e+62) {
		tmp = x * z;
	} else if (z <= 12000000000.0) {
		tmp = y * t;
	} else {
		tmp = x * z;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -5.5e+62:
		tmp = x * z
	elif z <= 12000000000.0:
		tmp = y * t
	else:
		tmp = x * z
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -5.5e+62)
		tmp = Float64(x * z);
	elseif (z <= 12000000000.0)
		tmp = Float64(y * t);
	else
		tmp = Float64(x * z);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -5.5e+62)
		tmp = x * z;
	elseif (z <= 12000000000.0)
		tmp = y * t;
	else
		tmp = x * z;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -5.5e+62], N[(x * z), $MachinePrecision], If[LessEqual[z, 12000000000.0], N[(y * t), $MachinePrecision], N[(x * z), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.4999999999999997e62 or 1.2e10 < 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. *-lowering-*.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. --lowering--.f64 82.2

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

   5. Simplified 82.2%

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

   6. Taylor expanded in x around inf

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

   8. Simplified 46.6%

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

   if -5.4999999999999997e62 < z < 1.2e10

   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. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, y - z, 0\right)} \]

      3. --lowering--.f64 47.1

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{y - z}, 0\right) \]

   5. Simplified 47.1%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, y - z, 0\right)} \]

   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. *-lowering-*.f64 36.7

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

   8. Simplified 36.7%

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

4. Final simplification 41.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -5.5 \cdot 10^{+62}:\\ \;\;\;\;x \cdot z\\ \mathbf{elif}\;z \leq 12000000000:\\ \;\;\;\;y \cdot t\\ \mathbf{else}:\\ \;\;\;\;x \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 37.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{-20}:\\ \;\;\;\;y \cdot t\\ \mathbf{elif}\;y \leq 6.8 \cdot 10^{-71}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;y \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -1.1e-20) (* y t) (if (<= y 6.8e-71) x (* y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.1e-20) {
		tmp = y * t;
	} else if (y <= 6.8e-71) {
		tmp = x;
	} else {
		tmp = y * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (y <= (-1.1d-20)) then
        tmp = y * t
    else if (y <= 6.8d-71) then
        tmp = x
    else
        tmp = y * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -1.1e-20) {
		tmp = y * t;
	} else if (y <= 6.8e-71) {
		tmp = x;
	} else {
		tmp = y * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if y <= -1.1e-20:
		tmp = y * t
	elif y <= 6.8e-71:
		tmp = x
	else:
		tmp = y * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -1.1e-20)
		tmp = Float64(y * t);
	elseif (y <= 6.8e-71)
		tmp = x;
	else
		tmp = Float64(y * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (y <= -1.1e-20)
		tmp = y * t;
	elseif (y <= 6.8e-71)
		tmp = x;
	else
		tmp = y * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -1.1e-20], N[(y * t), $MachinePrecision], If[LessEqual[y, 6.8e-71], x, N[(y * t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -1.09999999999999995e-20 or 6.80000000000000007e-71 < 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. +-rgt-identity N/A

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

      2. accelerator-lowering-fma.f64 N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(t, y - z, 0\right)} \]

      3. --lowering--.f64 56.8

         \[\leadsto \mathsf{fma}\left(t, \color{blue}{y - z}, 0\right) \]

   5. Simplified 56.8%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t, y - z, 0\right)} \]

   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. *-lowering-*.f64 42.2

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

   8. Simplified 42.2%

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

   if -1.09999999999999995e-20 < y < 6.80000000000000007e-71

   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. accelerator-lowering-fma.f64 N/A

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

      3. --lowering--.f64 38.3

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

   5. Simplified 38.3%

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

   6. Taylor expanded in y around 0

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

   8. Simplified 33.0%

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

Alternative 11: 18.6% accurate, 15.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_] := x

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 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. accelerator-lowering-fma.f64 N/A

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

   3. --lowering--.f64 56.6

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

5. Simplified 56.6%

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

6. Taylor expanded in y around 0

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

8. Simplified 16.9%

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

Developer Target 1: 96.4% 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 2024195 
(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))))