Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 7.1s

Alternatives: 11

Speedup: 1.2×

• 34.9% 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

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. lower-fma.f64 100.0

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

4. Applied rewrites 100.0%

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

Alternative 2: 69.7% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ t_2 := \left(y - z\right) \cdot t\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.7 \cdot 10^{-52}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-219}:\\ \;\;\;\;\mathsf{fma}\left(-t, z, x\right)\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+61}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)) (t_2 (* (- y z) t)))
   (if (<= y -1.05e+89)
     t_1
     (if (<= y -3.7e-52)
       t_2
       (if (<= y 8.8e-219)
         (fma (- t) z x)
         (if (<= y 1.5e-98) (fma x z x) (if (<= y 4.9e+61) t_2 t_1)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double t_2 = (y - z) * t;
	double tmp;
	if (y <= -1.05e+89) {
		tmp = t_1;
	} else if (y <= -3.7e-52) {
		tmp = t_2;
	} else if (y <= 8.8e-219) {
		tmp = fma(-t, z, x);
	} else if (y <= 1.5e-98) {
		tmp = fma(x, z, x);
	} else if (y <= 4.9e+61) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	t_2 = Float64(Float64(y - z) * t)
	tmp = 0.0
	if (y <= -1.05e+89)
		tmp = t_1;
	elseif (y <= -3.7e-52)
		tmp = t_2;
	elseif (y <= 8.8e-219)
		tmp = fma(Float64(-t), z, x);
	elseif (y <= 1.5e-98)
		tmp = fma(x, z, x);
	elseif (y <= 4.9e+61)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[y, -1.05e+89], t$95$1, If[LessEqual[y, -3.7e-52], t$95$2, If[LessEqual[y, 8.8e-219], N[((-t) * z + x), $MachinePrecision], If[LessEqual[y, 1.5e-98], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 4.9e+61], t$95$2, t$95$1]]]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -1.04999999999999993e89 or 4.90000000000000025e61 < 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 90.5

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

   5. Applied rewrites 90.5%

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

   if -1.04999999999999993e89 < y < -3.6999999999999997e-52 or 1.5e-98 < y < 4.90000000000000025e61

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 62.8

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

   5. Applied rewrites 62.8%

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

   if -3.6999999999999997e-52 < y < 8.7999999999999998e-219

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 96.2

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 74.3%

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

   if 8.7999999999999998e-219 < y < 1.5e-98

   1. Initial program 99.9%

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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 83.1

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

   5. Applied rewrites 83.1%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 83.1%

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

Alternative 3: 66.6% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ t_2 := \left(y - z\right) \cdot t\\ \mathbf{if}\;y \leq -1.05 \cdot 10^{+89}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3.1 \cdot 10^{-117}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 1.5 \cdot 10^{-98}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 4.9 \cdot 10^{+61}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)) (t_2 (* (- y z) t)))
   (if (<= y -1.05e+89)
     t_1
     (if (<= y -3.1e-117)
       t_2
       (if (<= y 1.5e-98) (fma x z x) (if (<= y 4.9e+61) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double t_2 = (y - z) * t;
	double tmp;
	if (y <= -1.05e+89) {
		tmp = t_1;
	} else if (y <= -3.1e-117) {
		tmp = t_2;
	} else if (y <= 1.5e-98) {
		tmp = fma(x, z, x);
	} else if (y <= 4.9e+61) {
		tmp = t_2;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	t_2 = Float64(Float64(y - z) * t)
	tmp = 0.0
	if (y <= -1.05e+89)
		tmp = t_1;
	elseif (y <= -3.1e-117)
		tmp = t_2;
	elseif (y <= 1.5e-98)
		tmp = fma(x, z, x);
	elseif (y <= 4.9e+61)
		tmp = t_2;
	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]}, Block[{t$95$2 = N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision]}, If[LessEqual[y, -1.05e+89], t$95$1, If[LessEqual[y, -3.1e-117], t$95$2, If[LessEqual[y, 1.5e-98], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 4.9e+61], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.04999999999999993e89 or 4.90000000000000025e61 < 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 90.5

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

   5. Applied rewrites 90.5%

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

   if -1.04999999999999993e89 < y < -3.10000000000000011e-117 or 1.5e-98 < y < 4.90000000000000025e61

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 62.3

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

   5. Applied rewrites 62.3%

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

   if -3.10000000000000011e-117 < y < 1.5e-98

   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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 62.8

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

   5. Applied rewrites 62.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 62.8%

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

Alternative 4: 83.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ \mathbf{if}\;z \leq -4.9 \cdot 10^{+71}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{-22}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)))
   (if (<= z -4.9e+71)
     t_1
     (if (<= z -8.6e-22)
       (* (- y z) t)
       (if (<= z 1.05e+86) (fma (- t x) y x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x - t) * z;
	double tmp;
	if (z <= -4.9e+71) {
		tmp = t_1;
	} else if (z <= -8.6e-22) {
		tmp = (y - z) * t;
	} else if (z <= 1.05e+86) {
		tmp = fma((t - x), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x - t) * z)
	tmp = 0.0
	if (z <= -4.9e+71)
		tmp = t_1;
	elseif (z <= -8.6e-22)
		tmp = Float64(Float64(y - z) * t);
	elseif (z <= 1.05e+86)
		tmp = fma(Float64(t - x), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[z, -4.9e+71], t$95$1, If[LessEqual[z, -8.6e-22], N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision], If[LessEqual[z, 1.05e+86], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.8999999999999997e71 or 1.0499999999999999e86 < 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

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower-/.f64 99.9

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

   5. Applied rewrites 99.9%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 99.9

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

   7. Applied rewrites 99.9%

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

   8. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. distribute-lft-out-- N/A

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

      5. mul-1-neg N/A

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

      6. fp-cancel-sub-sign-inv N/A

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

      7. metadata-eval N/A

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

      8. *-lft-identity N/A

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

      9. +-commutative N/A

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

      10. mul-1-neg N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. *-lft-identity N/A

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

      14. lower--.f64 85.2

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

   10. Applied rewrites 85.2%

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

   if -4.8999999999999997e71 < z < -8.60000000000000075e-22

   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. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 92.4

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

   5. Applied rewrites 92.4%

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

   if -8.60000000000000075e-22 < z < 1.0499999999999999e86

   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 86.7

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

   5. Applied rewrites 86.7%

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

4. Final simplification 86.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4.9 \cdot 10^{+71}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{elif}\;z \leq -8.6 \cdot 10^{-22}:\\ \;\;\;\;\left(y - z\right) \cdot t\\ \mathbf{elif}\;z \leq 1.05 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 48.2% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot y\\ \mathbf{if}\;y \leq -3.2 \cdot 10^{+168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -4.8 \cdot 10^{-59}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{+52}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x) y)))
   (if (<= y -3.2e+168)
     t_1
     (if (<= y -4.8e-59) (* t y) (if (<= y 1.7e+52) (fma x z x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -x * y;
	double tmp;
	if (y <= -3.2e+168) {
		tmp = t_1;
	} else if (y <= -4.8e-59) {
		tmp = t * y;
	} else if (y <= 1.7e+52) {
		tmp = fma(x, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-x) * y)
	tmp = 0.0
	if (y <= -3.2e+168)
		tmp = t_1;
	elseif (y <= -4.8e-59)
		tmp = Float64(t * y);
	elseif (y <= 1.7e+52)
		tmp = fma(x, z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[((-x) * y), $MachinePrecision]}, If[LessEqual[y, -3.2e+168], t$95$1, If[LessEqual[y, -4.8e-59], N[(t * y), $MachinePrecision], If[LessEqual[y, 1.7e+52], N[(x * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -3.2000000000000001e168 or 1.7e52 < 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 90.2

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

   5. Applied rewrites 90.2%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 65.8%

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

   if -3.2000000000000001e168 < y < -4.8000000000000003e-59

   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 67.7

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

   5. Applied rewrites 67.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 54.5%

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

   if -4.8000000000000003e-59 < y < 1.7e52

   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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 56.8

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

   5. Applied rewrites 56.8%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 52.8%

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

Alternative 6: 46.5% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -30000000000:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;x \leq 7.5 \cdot 10^{-245}:\\ \;\;\;\;\left(-t\right) \cdot z\\ \mathbf{elif}\;x \leq 5.4 \cdot 10^{-83}:\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -30000000000.0)
   (fma x z x)
   (if (<= x 7.5e-245) (* (- t) z) (if (<= x 5.4e-83) (* t y) (fma x z x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -30000000000.0) {
		tmp = fma(x, z, x);
	} else if (x <= 7.5e-245) {
		tmp = -t * z;
	} else if (x <= 5.4e-83) {
		tmp = t * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -30000000000.0)
		tmp = fma(x, z, x);
	elseif (x <= 7.5e-245)
		tmp = Float64(Float64(-t) * z);
	elseif (x <= 5.4e-83)
		tmp = Float64(t * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -30000000000.0], N[(x * z + x), $MachinePrecision], If[LessEqual[x, 7.5e-245], N[((-t) * z), $MachinePrecision], If[LessEqual[x, 5.4e-83], N[(t * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if x < -3e10 or 5.39999999999999982e-83 < 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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 82.7

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

   5. Applied rewrites 82.7%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 56.0%

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

   if -3e10 < x < 7.5000000000000003e-245

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.3%

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

   if 7.5000000000000003e-245 < x < 5.39999999999999982e-83

   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 62.7

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

   5. Applied rewrites 62.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 53.0%

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

Alternative 7: 81.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-54} \lor \neg \left(y \leq 3.9 \cdot 10^{-35}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -2.05e-54) (not (<= y 3.9e-35)))
   (fma (- t x) y x)
   (fma (- x t) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -2.05e-54) || !(y <= 3.9e-35)) {
		tmp = fma((t - x), y, x);
	} else {
		tmp = fma((x - t), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -2.05e-54) || !(y <= 3.9e-35))
		tmp = fma(Float64(t - x), y, x);
	else
		tmp = fma(Float64(x - t), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -2.05e-54], N[Not[LessEqual[y, 3.9e-35]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -2.05e-54 or 3.8999999999999998e-35 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 81.2

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

   5. Applied rewrites 81.2%

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

   if -2.05e-54 < y < 3.8999999999999998e-35

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. fp-cancel-sign-sub-inv N/A

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

      12. metadata-eval N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 95.7

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

   5. Applied rewrites 95.7%

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

4. Final simplification 87.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{-54} \lor \neg \left(y \leq 3.9 \cdot 10^{-35}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 65.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-59} \lor \neg \left(y \leq 2.1 \cdot 10^{-31}\right):\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4.8e-59) (not (<= y 2.1e-31))) (* (- t x) y) (fma x z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.8e-59) || !(y <= 2.1e-31)) {
		tmp = (t - x) * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4.8e-59) || !(y <= 2.1e-31))
		tmp = Float64(Float64(t - x) * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4.8e-59], N[Not[LessEqual[y, 2.1e-31]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000003e-59 or 2.09999999999999991e-31 < 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 74.6

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

   5. Applied rewrites 74.6%

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

   if -4.8000000000000003e-59 < y < 2.09999999999999991e-31

   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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.6%

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

4. Final simplification 67.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-59} \lor \neg \left(y \leq 2.1 \cdot 10^{-31}\right):\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-59} \lor \neg \left(y \leq 2.1 \cdot 10^{-31}\right):\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -4.8e-59) (not (<= y 2.1e-31))) (* t y) (fma x z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -4.8e-59) || !(y <= 2.1e-31)) {
		tmp = t * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -4.8e-59) || !(y <= 2.1e-31))
		tmp = Float64(t * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -4.8e-59], N[Not[LessEqual[y, 2.1e-31]], $MachinePrecision]], N[(t * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -4.8000000000000003e-59 or 2.09999999999999991e-31 < 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 74.6

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

   5. Applied rewrites 74.6%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 39.6%

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

   if -4.8000000000000003e-59 < y < 2.09999999999999991e-31

   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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 58.6

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

   5. Applied rewrites 58.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 58.6%

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

4. Final simplification 47.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -4.8 \cdot 10^{-59} \lor \neg \left(y \leq 2.1 \cdot 10^{-31}\right):\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 39.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+88} \lor \neg \left(z \leq 3.1 \cdot 10^{+37}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -3.4e+88) (not (<= z 3.1e+37))) (* x z) (* t y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.4e+88) || !(z <= 3.1e+37)) {
		tmp = x * z;
	} else {
		tmp = t * y;
	}
	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 <= (-3.4d+88)) .or. (.not. (z <= 3.1d+37))) then
        tmp = x * z
    else
        tmp = t * y
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -3.4e+88) || !(z <= 3.1e+37)) {
		tmp = x * z;
	} else {
		tmp = t * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (z <= -3.4e+88) or not (z <= 3.1e+37):
		tmp = x * z
	else:
		tmp = t * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -3.4e+88) || !(z <= 3.1e+37))
		tmp = Float64(x * z);
	else
		tmp = Float64(t * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((z <= -3.4e+88) || ~((z <= 3.1e+37)))
		tmp = x * z;
	else
		tmp = t * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -3.4e+88], N[Not[LessEqual[z, 3.1e+37]], $MachinePrecision]], N[(x * z), $MachinePrecision], N[(t * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -3.40000000000000004e88 or 3.1000000000000002e37 < z

   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

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

      2. +-commutative N/A

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

      3. distribute-lft1-in N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. *-lft-identity N/A

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

      7. metadata-eval N/A

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

      8. fp-cancel-sign-sub-inv N/A

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

      9. +-commutative N/A

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

      10. *-lft-identity N/A

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

      11. metadata-eval N/A

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

      12. fp-cancel-sub-sign-inv N/A

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

      13. distribute-lft-out-- N/A

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

      14. distribute-lft-neg-in N/A

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

      15. metadata-eval N/A

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

      16. distribute-lft-out-- N/A

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

      17. *-lft-identity N/A

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

      18. *-lft-identity N/A

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

      19. lower--.f64 55.4

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

   5. Applied rewrites 55.4%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 48.2%

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

   if -3.40000000000000004e88 < z < 3.1000000000000002e37

   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 55.5

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

   5. Applied rewrites 55.5%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 33.2%

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

4. Final simplification 38.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -3.4 \cdot 10^{+88} \lor \neg \left(z \leq 3.1 \cdot 10^{+37}\right):\\ \;\;\;\;x \cdot z\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.6% accurate, 2.5× speedup

Math

\[\begin{array}{l} \\ t \cdot y \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

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 45.4

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

5. Applied rewrites 45.4%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 25.6%

   \[\leadsto t \cdot \color{blue}{y} \]
9. Add Preprocessing

Developer Target 1: 96.3% 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 2024338 
(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))))