Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 6.5s

Alternatives: 11

Speedup: 1.0×

• 34.7% 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 99.9%

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

Alternative 2: 67.8% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ t_2 := \left(x - t\right) \cdot z\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -8.5 \cdot 10^{-167}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;y \leq 3.1 \cdot 10^{-261}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 7.5 \cdot 10^{+28}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)) (t_2 (* (- x t) z)))
   (if (<= y -8.5e+92)
     t_1
     (if (<= y -8.5e-167)
       t_2
       (if (<= y 3.1e-261) (fma x z x) (if (<= y 7.5e+28) t_2 t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double t_2 = (x - t) * z;
	double tmp;
	if (y <= -8.5e+92) {
		tmp = t_1;
	} else if (y <= -8.5e-167) {
		tmp = t_2;
	} else if (y <= 3.1e-261) {
		tmp = fma(x, z, x);
	} else if (y <= 7.5e+28) {
		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(x - t) * z)
	tmp = 0.0
	if (y <= -8.5e+92)
		tmp = t_1;
	elseif (y <= -8.5e-167)
		tmp = t_2;
	elseif (y <= 3.1e-261)
		tmp = fma(x, z, x);
	elseif (y <= 7.5e+28)
		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[(x - t), $MachinePrecision] * z), $MachinePrecision]}, If[LessEqual[y, -8.5e+92], t$95$1, If[LessEqual[y, -8.5e-167], t$95$2, If[LessEqual[y, 3.1e-261], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 7.5e+28], t$95$2, t$95$1]]]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5000000000000001e92 or 7.4999999999999998e28 < y

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 89.3

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

   5. Applied rewrites 89.3%

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

   if -8.5000000000000001e92 < y < -8.4999999999999994e-167 or 3.0999999999999998e-261 < y < 7.4999999999999998e28

   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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

   5. Applied rewrites 89.3%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
   7. 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. fp-cancel-sub-sign-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 63.8

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

   8. Applied rewrites 63.8%

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

   if -8.4999999999999994e-167 < y < 3.0999999999999998e-261

   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 97.9

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

   5. Applied rewrites 97.9%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 76.9%

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

Alternative 3: 71.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -8.5 \cdot 10^{+92}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -1.85 \cdot 10^{-41}:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{elif}\;y \leq 900000000:\\ \;\;\;\;\mathsf{fma}\left(-t, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -8.5e+92)
     t_1
     (if (<= y -1.85e-41)
       (* (- x t) z)
       (if (<= y 900000000.0) (fma (- t) z x) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -8.5e+92) {
		tmp = t_1;
	} else if (y <= -1.85e-41) {
		tmp = (x - t) * z;
	} else if (y <= 900000000.0) {
		tmp = fma(-t, z, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	tmp = 0.0
	if (y <= -8.5e+92)
		tmp = t_1;
	elseif (y <= -1.85e-41)
		tmp = Float64(Float64(x - t) * z);
	elseif (y <= 900000000.0)
		tmp = fma(Float64(-t), z, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -8.5e+92], t$95$1, If[LessEqual[y, -1.85e-41], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], If[LessEqual[y, 900000000.0], N[((-t) * z + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -8.5000000000000001e92 or 9e8 < y

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 87.8

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

   5. Applied rewrites 87.8%

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

   if -8.5000000000000001e92 < y < -1.8500000000000001e-41

   1. Initial program 100.0%

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

   3. Taylor expanded in x around inf

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
   7. 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. fp-cancel-sub-sign-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 62.3

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

   8. Applied rewrites 62.3%

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

   if -1.8500000000000001e-41 < y < 9e8

   1. Initial program 99.9%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-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 94.0

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

   5. Applied rewrites 94.0%

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

   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 71.5%

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

Alternative 4: 67.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -1550000000000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 900000000:\\ \;\;\;\;\left(-z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -1550000000000.0)
     t_1
     (if (<= y 1.7e-32) (fma x z x) (if (<= y 900000000.0) (* (- z) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -1550000000000.0) {
		tmp = t_1;
	} else if (y <= 1.7e-32) {
		tmp = fma(x, z, x);
	} else if (y <= 900000000.0) {
		tmp = -z * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	tmp = 0.0
	if (y <= -1550000000000.0)
		tmp = t_1;
	elseif (y <= 1.7e-32)
		tmp = fma(x, z, x);
	elseif (y <= 900000000.0)
		tmp = Float64(Float64(-z) * t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -1550000000000.0], t$95$1, If[LessEqual[y, 1.7e-32], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 900000000.0], N[((-z) * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.55e12 or 9e8 < y

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 79.7

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

   5. Applied rewrites 79.7%

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

   if -1.55e12 < y < 1.69999999999999989e-32

   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 91.6

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

   5. Applied rewrites 91.6%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 63.1%

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

   if 1.69999999999999989e-32 < y < 9e8

   1. Initial program 99.8%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 78.0

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

   5. Applied rewrites 78.0%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 78.1%

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

Alternative 5: 49.3% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(-x\right) \cdot y\\ \mathbf{if}\;y \leq -1.1 \cdot 10^{+93}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.7 \cdot 10^{-32}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 1.6 \cdot 10^{+50}:\\ \;\;\;\;\left(-z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x) y)))
   (if (<= y -1.1e+93)
     t_1
     (if (<= y 1.7e-32) (fma x z x) (if (<= y 1.6e+50) (* (- z) t) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = -x * y;
	double tmp;
	if (y <= -1.1e+93) {
		tmp = t_1;
	} else if (y <= 1.7e-32) {
		tmp = fma(x, z, x);
	} else if (y <= 1.6e+50) {
		tmp = -z * t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(-x) * y)
	tmp = 0.0
	if (y <= -1.1e+93)
		tmp = t_1;
	elseif (y <= 1.7e-32)
		tmp = fma(x, z, x);
	elseif (y <= 1.6e+50)
		tmp = Float64(Float64(-z) * t);
	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, -1.1e+93], t$95$1, If[LessEqual[y, 1.7e-32], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 1.6e+50], N[((-z) * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.10000000000000011e93 or 1.59999999999999991e50 < y

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 91.5

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

   5. Applied rewrites 91.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 54.1%

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

   if -1.10000000000000011e93 < y < 1.69999999999999989e-32

   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 88.1

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

   5. Applied rewrites 88.1%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 59.1%

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

   if 1.69999999999999989e-32 < y < 1.59999999999999991e50

   1. Initial program 99.9%

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

   3. Taylor expanded in x around 0

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

      1. *-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 70.6

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

   5. Applied rewrites 70.6%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 50.5%

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

Alternative 6: 82.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+92} \lor \neg \left(y \leq 7.5 \cdot 10^{+28}\right):\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -8.5e+92) (not (<= y 7.5e+28)))
   (* (- t x) y)
   (fma (- x t) z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -8.5e+92) || !(y <= 7.5e+28)) {
		tmp = (t - x) * y;
	} else {
		tmp = fma((x - t), z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -8.5e+92) || !(y <= 7.5e+28))
		tmp = Float64(Float64(t - x) * y);
	else
		tmp = fma(Float64(x - t), z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -8.5e+92], N[Not[LessEqual[y, 7.5e+28]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -8.5000000000000001e92 or 7.4999999999999998e28 < y

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 89.3

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

   5. Applied rewrites 89.3%

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

   if -8.5000000000000001e92 < y < 7.4999999999999998e28

   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 87.2

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

   5. Applied rewrites 87.2%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -8.5 \cdot 10^{+92} \lor \neg \left(y \leq 7.5 \cdot 10^{+28}\right):\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+43} \lor \neg \left(z \leq 365000000\right):\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= z -6.6e+43) (not (<= z 365000000.0)))
   (* (- x t) z)
   (fma (- t x) y x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((z <= -6.6e+43) || !(z <= 365000000.0)) {
		tmp = (x - t) * z;
	} else {
		tmp = fma((t - x), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((z <= -6.6e+43) || !(z <= 365000000.0))
		tmp = Float64(Float64(x - t) * z);
	else
		tmp = fma(Float64(t - x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[z, -6.6e+43], N[Not[LessEqual[z, 365000000.0]], $MachinePrecision]], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.6000000000000003e43 or 3.65e8 < 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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

   5. Applied rewrites 96.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
   7. 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. fp-cancel-sub-sign-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 80.8

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

   8. Applied rewrites 80.8%

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

   if -6.6000000000000003e43 < z < 3.65e8

   1. Initial program 99.9%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. lower--.f64 87.0

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

   5. Applied rewrites 87.0%

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

4. Final simplification 84.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.6 \cdot 10^{+43} \lor \neg \left(z \leq 365000000\right):\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 49.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+93} \lor \neg \left(y \leq 31\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -1.1e+93) (not (<= y 31.0))) (* (- x) y) (fma x z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -1.1e+93) || !(y <= 31.0)) {
		tmp = -x * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -1.1e+93) || !(y <= 31.0))
		tmp = Float64(Float64(-x) * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -1.1e+93], N[Not[LessEqual[y, 31.0]], $MachinePrecision]], N[((-x) * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.10000000000000011e93 or 31 < y

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 84.7

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

   5. Applied rewrites 84.7%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 48.9%

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

   if -1.10000000000000011e93 < y < 31

   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 88.5

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

   5. Applied rewrites 88.5%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 58.0%

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

4. Final simplification 54.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -1.1 \cdot 10^{+93} \lor \neg \left(y \leq 31\right):\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 48.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+109} \lor \neg \left(y \leq 8.5 \cdot 10^{+102}\right):\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= y -3.5e+109) (not (<= y 8.5e+102))) (* t y) (fma x z x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((y <= -3.5e+109) || !(y <= 8.5e+102)) {
		tmp = t * y;
	} else {
		tmp = fma(x, z, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((y <= -3.5e+109) || !(y <= 8.5e+102))
		tmp = Float64(t * y);
	else
		tmp = fma(x, z, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[y, -3.5e+109], N[Not[LessEqual[y, 8.5e+102]], $MachinePrecision]], N[(t * y), $MachinePrecision], N[(x * z + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -3.49999999999999983e109 or 8.4999999999999996e102 < y

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 93.7

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

   5. Applied rewrites 93.7%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 45.5%

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

   if -3.49999999999999983e109 < y < 8.4999999999999996e102

   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 83.4

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

   5. Applied rewrites 83.4%

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

   6. Taylor expanded in x around inf

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

   8. Applied rewrites 53.3%

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

4. Final simplification 50.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+109} \lor \neg \left(y \leq 8.5 \cdot 10^{+102}\right):\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 34.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-8} \lor \neg \left(x \leq 4.2 \cdot 10^{-69}\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (or (<= x -1e-8) (not (<= x 4.2e-69))) (* z x) (* t y)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x <= -1e-8) || !(x <= 4.2e-69)) {
		tmp = z * x;
	} 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 ((x <= (-1d-8)) .or. (.not. (x <= 4.2d-69))) then
        tmp = z * x
    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 ((x <= -1e-8) || !(x <= 4.2e-69)) {
		tmp = z * x;
	} else {
		tmp = t * y;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if (x <= -1e-8) or not (x <= 4.2e-69):
		tmp = z * x
	else:
		tmp = t * y
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if ((x <= -1e-8) || !(x <= 4.2e-69))
		tmp = Float64(z * x);
	else
		tmp = Float64(t * y);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if ((x <= -1e-8) || ~((x <= 4.2e-69)))
		tmp = z * x;
	else
		tmp = t * y;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Or[LessEqual[x, -1e-8], N[Not[LessEqual[x, 4.2e-69]], $MachinePrecision]], N[(z * x), $MachinePrecision], N[(t * y), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < -1e-8 or 4.1999999999999999e-69 < 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

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

      1. +-commutative N/A

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

      2. distribute-lft-in N/A

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

   5. Applied rewrites 99.2%

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

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{-1 \cdot \left(z \cdot \left(t - x\right)\right)} \]
   7. 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. fp-cancel-sub-sign-inv N/A

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

      6. metadata-eval N/A

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

      7. *-lft-identity N/A

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

      8. +-commutative N/A

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

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

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

      10. metadata-eval N/A

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

      11. *-lft-identity N/A

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

      12. lower--.f64 46.5

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

   8. Applied rewrites 46.5%

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

   9. Taylor expanded in x around inf

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

   11. Applied rewrites 36.1%

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

   if -1e-8 < x < 4.1999999999999999e-69

   1. Initial program 99.9%

      \[x + \left(y - z\right) \cdot \left(t - x\right) \]
   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 43.0

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

   5. Applied rewrites 43.0%

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

   6. Taylor expanded in x around 0

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

   8. Applied rewrites 34.8%

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

4. Final simplification 35.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -1 \cdot 10^{-8} \lor \neg \left(x \leq 4.2 \cdot 10^{-69}\right):\\ \;\;\;\;z \cdot x\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 26.5% 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 99.9%

   \[x + \left(y - z\right) \cdot \left(t - x\right) \]
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 41.3

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

5. Applied rewrites 41.3%

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

6. Taylor expanded in x around 0

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

8. Applied rewrites 21.5%

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

Developer Target 1: 96.2% 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 2024337 
(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))))