Data.Metrics.Snapshot:quantile from metrics-0.3.0.2

Percentage Accurate: 100.0% → 100.0%

Time: 8.4s

Alternatives: 12

Speedup: 1.0×

• 34.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 100.0% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 100.0%

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

3. Final simplification 100.0%

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

Alternative 2: 68.3% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -3.5 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-106}:\\ \;\;\;\;t \cdot \left(y - z\right)\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-280}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 45:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -3.5e+15)
     t_1
     (if (<= y -3e-106)
       (* t (- y z))
       (if (<= y -1.4e-280) (fma x z x) (if (<= y 45.0) (* (- x t) z) t_1))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -3.5e+15) {
		tmp = t_1;
	} else if (y <= -3e-106) {
		tmp = t * (y - z);
	} else if (y <= -1.4e-280) {
		tmp = fma(x, z, x);
	} else if (y <= 45.0) {
		tmp = (x - t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(t - x) * y)
	tmp = 0.0
	if (y <= -3.5e+15)
		tmp = t_1;
	elseif (y <= -3e-106)
		tmp = Float64(t * Float64(y - z));
	elseif (y <= -1.4e-280)
		tmp = fma(x, z, x);
	elseif (y <= 45.0)
		tmp = Float64(Float64(x - t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -3.5e+15], t$95$1, If[LessEqual[y, -3e-106], N[(t * N[(y - z), $MachinePrecision]), $MachinePrecision], If[LessEqual[y, -1.4e-280], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 45.0], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -3.5e15 or 45 < 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 88.7

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

   5. Applied rewrites 88.7%

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

   if -3.5e15 < y < -3.00000000000000019e-106

   1. Initial program 99.9%

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

   3. Taylor expanded in t around inf

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

      1. *-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 64.7

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

   5. Applied rewrites 64.7%

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

   if -3.00000000000000019e-106 < y < -1.40000000000000009e-280

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 92.9

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

   5. Applied rewrites 92.9%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 73.5%

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

   if -1.40000000000000009e-280 < y < 45

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 76.2

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

   5. Applied rewrites 76.2%

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

4. Final simplification 81.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -3.5 \cdot 10^{+15}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{elif}\;y \leq -3 \cdot 10^{-106}:\\ \;\;\;\;t \cdot \left(y - z\right)\\ \mathbf{elif}\;y \leq -1.4 \cdot 10^{-280}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 45:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 47.9% accurate, 0.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+45}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 29.5:\\ \;\;\;\;\left(-z\right) \cdot t\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+157}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.4e+45)
   (* t y)
   (if (<= y 1.85e-149)
     (fma x z x)
     (if (<= y 29.5) (* (- z) t) (if (<= y 5.2e+157) (* (- x) y) (* t y))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.4e+45) {
		tmp = t * y;
	} else if (y <= 1.85e-149) {
		tmp = fma(x, z, x);
	} else if (y <= 29.5) {
		tmp = -z * t;
	} else if (y <= 5.2e+157) {
		tmp = -x * y;
	} else {
		tmp = t * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.4e+45)
		tmp = Float64(t * y);
	elseif (y <= 1.85e-149)
		tmp = fma(x, z, x);
	elseif (y <= 29.5)
		tmp = Float64(Float64(-z) * t);
	elseif (y <= 5.2e+157)
		tmp = Float64(Float64(-x) * y);
	else
		tmp = Float64(t * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.4e+45], N[(t * y), $MachinePrecision], If[LessEqual[y, 1.85e-149], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 29.5], N[((-z) * t), $MachinePrecision], If[LessEqual[y, 5.2e+157], N[((-x) * y), $MachinePrecision], N[(t * y), $MachinePrecision]]]]]

Derivation

1. Split input into 4 regimes

2. if y < -7.39999999999999954e45 or 5.20000000000000022e157 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-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 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 57.4%

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

   if -7.39999999999999954e45 < y < 1.84999999999999995e-149

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 87.2

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

   5. Applied rewrites 87.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 57.8%

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

   if 1.84999999999999995e-149 < y < 29.5

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-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 60.0

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

   5. Applied rewrites 60.0%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 52.9%

      \[\leadsto \left(-z\right) \cdot t \]

   if 29.5 < y < 5.20000000000000022e157

   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 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.0%

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

Alternative 4: 70.4% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -7.8 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 6.2 \cdot 10^{-269}:\\ \;\;\;\;\mathsf{fma}\left(-t, z, x\right)\\ \mathbf{elif}\;y \leq 45:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -7.8e+14)
     t_1
     (if (<= y 6.2e-269) (fma (- t) z x) (if (<= y 45.0) (* (- x t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -7.8e+14) {
		tmp = t_1;
	} else if (y <= 6.2e-269) {
		tmp = fma(-t, z, x);
	} else if (y <= 45.0) {
		tmp = (x - t) * z;
	} 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 <= -7.8e+14)
		tmp = t_1;
	elseif (y <= 6.2e-269)
		tmp = fma(Float64(-t), z, x);
	elseif (y <= 45.0)
		tmp = Float64(Float64(x - t) * z);
	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, -7.8e+14], t$95$1, If[LessEqual[y, 6.2e-269], N[((-t) * z + x), $MachinePrecision], If[LessEqual[y, 45.0], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.8e14 or 45 < 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 88.7

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

   5. Applied rewrites 88.7%

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

   if -7.8e14 < y < 6.19999999999999933e-269

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 84.9

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

   5. Applied rewrites 84.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 66.4%

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

   if 6.19999999999999933e-269 < y < 45

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 79.6

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

   5. Applied rewrites 79.6%

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

Alternative 5: 64.9% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -1.85 \cdot 10^{-23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 1.85 \cdot 10^{-149}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 1.3 \cdot 10^{-14}:\\ \;\;\;\;\left(-z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -1.85e-23)
     t_1
     (if (<= y 1.85e-149) (fma x z x) (if (<= y 1.3e-14) (* (- 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 <= -1.85e-23) {
		tmp = t_1;
	} else if (y <= 1.85e-149) {
		tmp = fma(x, z, x);
	} else if (y <= 1.3e-14) {
		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 <= -1.85e-23)
		tmp = t_1;
	elseif (y <= 1.85e-149)
		tmp = fma(x, z, x);
	elseif (y <= 1.3e-14)
		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, -1.85e-23], t$95$1, If[LessEqual[y, 1.85e-149], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 1.3e-14], N[((-z) * t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if y < -1.8500000000000001e-23 or 1.29999999999999998e-14 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in y around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 83.4

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

   5. Applied rewrites 83.4%

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

   if -1.8500000000000001e-23 < y < 1.84999999999999995e-149

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 92.2

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

   5. Applied rewrites 92.2%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 61.4%

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

   if 1.84999999999999995e-149 < y < 1.29999999999999998e-14

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-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 63.1

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

   5. Applied rewrites 63.1%

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

   6. Taylor expanded in z around inf

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

   8. Applied rewrites 58.9%

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

Alternative 6: 50.6% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+45}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq 96:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{elif}\;y \leq 5.2 \cdot 10^{+157}:\\ \;\;\;\;\left(-x\right) \cdot y\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.4e+45)
   (* t y)
   (if (<= y 96.0) (fma x z x) (if (<= y 5.2e+157) (* (- x) y) (* t y)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.4e+45) {
		tmp = t * y;
	} else if (y <= 96.0) {
		tmp = fma(x, z, x);
	} else if (y <= 5.2e+157) {
		tmp = -x * y;
	} else {
		tmp = t * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.4e+45)
		tmp = Float64(t * y);
	elseif (y <= 96.0)
		tmp = fma(x, z, x);
	elseif (y <= 5.2e+157)
		tmp = Float64(Float64(-x) * y);
	else
		tmp = Float64(t * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.4e+45], N[(t * y), $MachinePrecision], If[LessEqual[y, 96.0], N[(x * z + x), $MachinePrecision], If[LessEqual[y, 5.2e+157], N[((-x) * y), $MachinePrecision], N[(t * y), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if y < -7.39999999999999954e45 or 5.20000000000000022e157 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-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 60.5

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

   5. Applied rewrites 60.5%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 57.4%

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

   if -7.39999999999999954e45 < y < 96

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 87.4

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

   5. Applied rewrites 87.4%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 53.6%

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

   if 96 < y < 5.20000000000000022e157

   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 79.1

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

   5. Applied rewrites 79.1%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 52.0%

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

Alternative 7: 84.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -2.7 \cdot 10^{+15}:\\ \;\;\;\;\left(t - x\right) \cdot y\\ \mathbf{elif}\;y \leq 8.8 \cdot 10^{-5}:\\ \;\;\;\;\mathsf{fma}\left(x - t, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -2.7e+15)
   (* (- t x) y)
   (if (<= y 8.8e-5) (fma (- x t) z x) (fma (- t x) y x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -2.7e+15) {
		tmp = (t - x) * y;
	} else if (y <= 8.8e-5) {
		tmp = fma((x - t), z, x);
	} else {
		tmp = fma((t - x), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -2.7e+15)
		tmp = Float64(Float64(t - x) * y);
	elseif (y <= 8.8e-5)
		tmp = fma(Float64(x - t), z, x);
	else
		tmp = fma(Float64(t - x), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -2.7e+15], N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision], If[LessEqual[y, 8.8e-5], N[(N[(x - t), $MachinePrecision] * z + x), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if y < -2.7e15

   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 -2.7e15 < y < 8.7999999999999998e-5

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 89.1

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

   5. Applied rewrites 89.1%

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

   if 8.7999999999999998e-5 < 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 88.4

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

   5. Applied rewrites 88.4%

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

Alternative 8: 85.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x - t\right) \cdot z\\ \mathbf{if}\;z \leq -1.4 \cdot 10^{+23}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2500000000:\\ \;\;\;\;\mathsf{fma}\left(t - x, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- x t) z)))
   (if (<= z -1.4e+23) t_1 (if (<= z 2500000000.0) (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 <= -1.4e+23) {
		tmp = t_1;
	} else if (z <= 2500000000.0) {
		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 <= -1.4e+23)
		tmp = t_1;
	elseif (z <= 2500000000.0)
		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, -1.4e+23], t$95$1, If[LessEqual[z, 2500000000.0], N[(N[(t - x), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4e23 or 2.5e9 < z

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 81.7

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

   5. Applied rewrites 81.7%

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

   if -1.4e23 < z < 2.5e9

   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 91.1

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

   5. Applied rewrites 91.1%

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

Alternative 9: 68.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - x\right) \cdot y\\ \mathbf{if}\;y \leq -2.7 \cdot 10^{+15}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 45:\\ \;\;\;\;\left(x - t\right) \cdot z\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- t x) y)))
   (if (<= y -2.7e+15) t_1 (if (<= y 45.0) (* (- x t) z) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -2.7e+15) {
		tmp = t_1;
	} else if (y <= 45.0) {
		tmp = (x - t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (t - x) * y
    if (y <= (-2.7d+15)) then
        tmp = t_1
    else if (y <= 45.0d0) then
        tmp = (x - t) * z
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (t - x) * y;
	double tmp;
	if (y <= -2.7e+15) {
		tmp = t_1;
	} else if (y <= 45.0) {
		tmp = (x - t) * z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (t - x) * y
	tmp = 0
	if y <= -2.7e+15:
		tmp = t_1
	elif y <= 45.0:
		tmp = (x - t) * z
	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 <= -2.7e+15)
		tmp = t_1;
	elseif (y <= 45.0)
		tmp = Float64(Float64(x - t) * z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (t - x) * y;
	tmp = 0.0;
	if (y <= -2.7e+15)
		tmp = t_1;
	elseif (y <= 45.0)
		tmp = (x - t) * z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(t - x), $MachinePrecision] * y), $MachinePrecision]}, If[LessEqual[y, -2.7e+15], t$95$1, If[LessEqual[y, 45.0], N[(N[(x - t), $MachinePrecision] * z), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.7e15 or 45 < 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 88.7

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

   5. Applied rewrites 88.7%

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

   if -2.7e15 < y < 45

   1. Initial program 100.0%

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

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-*r* N/A

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

      3. lower-*.f64 N/A

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

      4. mul-1-neg N/A

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

      5. sub-neg N/A

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

      6. +-commutative N/A

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

      7. distribute-neg-in N/A

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

      8. unsub-neg N/A

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

      9. remove-double-neg N/A

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

      10. lower--.f64 64.8

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

   5. Applied rewrites 64.8%

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

Alternative 10: 50.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;y \leq -7.4 \cdot 10^{+45}:\\ \;\;\;\;t \cdot y\\ \mathbf{elif}\;y \leq 8.2 \cdot 10^{+59}:\\ \;\;\;\;\mathsf{fma}\left(x, z, x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= y -7.4e+45) (* t y) (if (<= y 8.2e+59) (fma x z x) (* t y))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (y <= -7.4e+45) {
		tmp = t * y;
	} else if (y <= 8.2e+59) {
		tmp = fma(x, z, x);
	} else {
		tmp = t * y;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (y <= -7.4e+45)
		tmp = Float64(t * y);
	elseif (y <= 8.2e+59)
		tmp = fma(x, z, x);
	else
		tmp = Float64(t * y);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[y, -7.4e+45], N[(t * y), $MachinePrecision], If[LessEqual[y, 8.2e+59], N[(x * z + x), $MachinePrecision], N[(t * y), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if y < -7.39999999999999954e45 or 8.2e59 < y

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-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 57.1

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

   5. Applied rewrites 57.1%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 54.6%

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

   if -7.39999999999999954e45 < y < 8.2e59

   1. Initial program 100.0%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* N/A

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

      4. lower-fma.f64 N/A

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

      5. mul-1-neg N/A

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

      6. sub-neg N/A

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

      7. +-commutative N/A

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

      8. distribute-neg-in N/A

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

      9. unsub-neg N/A

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

      10. remove-double-neg N/A

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

      11. lower--.f64 83.6

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

   5. Applied rewrites 83.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 51.2%

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

Alternative 11: 40.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+16}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 2500000000:\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.4e+16) (* z x) (if (<= z 2500000000.0) (* t y) (* z x))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.4e+16) {
		tmp = z * x;
	} else if (z <= 2500000000.0) {
		tmp = t * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (z <= (-1.4d+16)) then
        tmp = z * x
    else if (z <= 2500000000.0d0) then
        tmp = t * y
    else
        tmp = z * x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.4e+16) {
		tmp = z * x;
	} else if (z <= 2500000000.0) {
		tmp = t * y;
	} else {
		tmp = z * x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if z <= -1.4e+16:
		tmp = z * x
	elif z <= 2500000000.0:
		tmp = t * y
	else:
		tmp = z * x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.4e+16)
		tmp = Float64(z * x);
	elseif (z <= 2500000000.0)
		tmp = Float64(t * y);
	else
		tmp = Float64(z * x);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (z <= -1.4e+16)
		tmp = z * x;
	elseif (z <= 2500000000.0)
		tmp = t * y;
	else
		tmp = z * x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.4e+16], N[(z * x), $MachinePrecision], If[LessEqual[z, 2500000000.0], N[(t * y), $MachinePrecision], N[(z * x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.4e16 or 2.5e9 < z

   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. lift--.f64 N/A

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

      5. sub-neg N/A

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

      6. distribute-lft-in N/A

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

      7. associate-+l+ N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-neg.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in t around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-*r* 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. sub-neg N/A

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

      7. mul-1-neg N/A

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

      8. +-commutative N/A

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

      9. distribute-neg-in N/A

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

      10. unsub-neg N/A

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

      11. mul-1-neg N/A

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

      12. remove-double-neg N/A

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

      13. lower--.f64 57.9

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

   7. Applied rewrites 57.9%

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

   8. Taylor expanded in z around inf

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

   10. Applied rewrites 44.7%

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

   if -1.4e16 < z < 2.5e9

   1. Initial program 100.0%

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

   3. Taylor expanded in t around inf

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

      1. *-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 51.3

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

   5. Applied rewrites 51.3%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 44.1%

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

4. Final simplification 44.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{+16}:\\ \;\;\;\;z \cdot x\\ \mathbf{elif}\;z \leq 2500000000:\\ \;\;\;\;t \cdot y\\ \mathbf{else}:\\ \;\;\;\;z \cdot x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 27.3% 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 t around inf

   \[\leadsto \color{blue}{t \cdot \left(y - z\right)} \]
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 50.8

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

5. Applied rewrites 50.8%

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

6. Taylor expanded in z around 0

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

8. Applied rewrites 31.1%

   \[\leadsto t \cdot \color{blue}{y} \]
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 2024276 
(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))))