Numeric.Histogram:binBounds from Chart-1.5.3

Percentage Accurate: 92.2% → 98.1%

Time: 10.1s

Alternatives: 10

Speedup: 1.1×

• 24.8% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

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 - x) * z) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 92.2% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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 - x) * z) / t)
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% accurate, 1.1× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 98.0

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

4. Applied egg-rr 98.0%

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

Alternative 2: 83.8% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.52 \cdot 10^{+122}:\\ \;\;\;\;z \cdot \frac{y - x}{t}\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z}{t} \cdot \left(y - x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= z -1.52e+122)
   (* z (/ (- y x) t))
   (if (<= z 2.7e+31) (fma (/ z t) y x) (* (/ z t) (- y x)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (z <= -1.52e+122) {
		tmp = z * ((y - x) / t);
	} else if (z <= 2.7e+31) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = (z / t) * (y - x);
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (z <= -1.52e+122)
		tmp = Float64(z * Float64(Float64(y - x) / t));
	elseif (z <= 2.7e+31)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(Float64(z / t) * Float64(y - x));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[z, -1.52e+122], N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.7e+31], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(z / t), $MachinePrecision] * N[(y - x), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.52e122

   1. Initial program 82.5%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      4. lower--.f64 92.9

         \[\leadsto z \cdot \frac{\color{blue}{y - x}}{t} \]

   5. Simplified 92.9%

      \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

   if -1.52e122 < z < 2.69999999999999986e31

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 89.1

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

   5. Simplified 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.2

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

   7. Applied egg-rr 90.2%

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

   if 2.69999999999999986e31 < z

   1. Initial program 79.1%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      4. lower--.f64 82.8

         \[\leadsto z \cdot \frac{\color{blue}{y - x}}{t} \]

   5. Simplified 82.8%

      \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]
   6. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto z \cdot \frac{\color{blue}{y - x}}{t} \]

      2. clear-num N/A

         \[\leadsto z \cdot \color{blue}{\frac{1}{\frac{t}{y - x}}} \]

      3. associate-/r/ N/A

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

      4. associate-*r* N/A

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

      5. div-inv N/A

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

      6. lift-/.f64 N/A

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

      7. lower-*.f64 86.4

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

   7. Applied egg-rr 86.4%

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

Alternative 3: 84.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := z \cdot \frac{y - x}{t}\\ \mathbf{if}\;z \leq -1.52 \cdot 10^{+122}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.7 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* z (/ (- y x) t))))
   (if (<= z -1.52e+122) t_1 (if (<= z 2.7e+31) (fma (/ z t) y x) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = z * ((y - x) / t);
	double tmp;
	if (z <= -1.52e+122) {
		tmp = t_1;
	} else if (z <= 2.7e+31) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(z * Float64(Float64(y - x) / t))
	tmp = 0.0
	if (z <= -1.52e+122)
		tmp = t_1;
	elseif (z <= 2.7e+31)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(z * N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[z, -1.52e+122], t$95$1, If[LessEqual[z, 2.7e+31], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.52e122 or 2.69999999999999986e31 < z

   1. Initial program 80.5%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      4. lower--.f64 86.9

         \[\leadsto z \cdot \frac{\color{blue}{y - x}}{t} \]

   5. Simplified 86.9%

      \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

   if -1.52e122 < z < 2.69999999999999986e31

   1. Initial program 96.8%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 89.1

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

   5. Simplified 89.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 90.2

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

   7. Applied egg-rr 90.2%

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

Alternative 4: 56.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.8 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 0.115:\\ \;\;\;\;\frac{z}{t} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.8e-29) x (if (<= t 0.115) (* (/ z t) y) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.8e-29) {
		tmp = x;
	} else if (t <= 0.115) {
		tmp = (z / t) * y;
	} else {
		tmp = 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 (t <= (-2.8d-29)) then
        tmp = x
    else if (t <= 0.115d0) then
        tmp = (z / t) * y
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.8e-29) {
		tmp = x;
	} else if (t <= 0.115) {
		tmp = (z / t) * y;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.8e-29:
		tmp = x
	elif t <= 0.115:
		tmp = (z / t) * y
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.8e-29)
		tmp = x;
	elseif (t <= 0.115)
		tmp = Float64(Float64(z / t) * y);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.8e-29)
		tmp = x;
	elseif (t <= 0.115)
		tmp = (z / t) * y;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.8e-29], x, If[LessEqual[t, 0.115], N[(N[(z / t), $MachinePrecision] * y), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -2.8000000000000002e-29 or 0.115000000000000005 < t

   1. Initial program 85.6%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 99.2

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

   4. Applied egg-rr 99.2%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

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

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

      3. unsub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 67.2

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

   7. Simplified 67.2%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 45.3

         \[\leadsto \frac{\color{blue}{t \cdot x}}{t} \]

   10. Simplified 45.3%

       \[\leadsto \frac{\color{blue}{t \cdot x}}{t} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot t}}{t} \]

       2. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{t}{t}} \]

       3. *-inverses N/A

          \[\leadsto x \cdot \color{blue}{1} \]

       4. metadata-eval N/A

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

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

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

       6. *-commutative N/A

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

       7. neg-mul-1 N/A

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

       8. remove-double-neg 65.5

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

   12. Applied egg-rr 65.5%

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

   if -2.8000000000000002e-29 < t < 0.115000000000000005

   1. Initial program 97.9%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 48.9

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

   5. Simplified 48.9%

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

      1. associate-*r/ N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lower-*.f64 55.4

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

   7. Applied egg-rr 55.4%

      \[\leadsto \color{blue}{\frac{z}{t} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 52.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -2.6 \cdot 10^{-29}:\\ \;\;\;\;x\\ \mathbf{elif}\;t \leq 8.5 \cdot 10^{+90}:\\ \;\;\;\;z \cdot \frac{y}{t}\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -2.6e-29) x (if (<= t 8.5e+90) (* z (/ y t)) x)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e-29) {
		tmp = x;
	} else if (t <= 8.5e+90) {
		tmp = z * (y / t);
	} else {
		tmp = 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 (t <= (-2.6d-29)) then
        tmp = x
    else if (t <= 8.5d+90) then
        tmp = z * (y / t)
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -2.6e-29) {
		tmp = x;
	} else if (t <= 8.5e+90) {
		tmp = z * (y / t);
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -2.6e-29:
		tmp = x
	elif t <= 8.5e+90:
		tmp = z * (y / t)
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -2.6e-29)
		tmp = x;
	elseif (t <= 8.5e+90)
		tmp = Float64(z * Float64(y / t));
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -2.6e-29)
		tmp = x;
	elseif (t <= 8.5e+90)
		tmp = z * (y / t);
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -2.6e-29], x, If[LessEqual[t, 8.5e+90], N[(z * N[(y / t), $MachinePrecision]), $MachinePrecision], x]]

Derivation

1. Split input into 2 regimes

2. if t < -2.6000000000000002e-29 or 8.5000000000000002e90 < t

   1. Initial program 84.8%

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

      1. lift--.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. lift-/.f64 N/A

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

      4. +-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-/l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 99.9

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

   4. Applied egg-rr 99.9%

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

   5. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

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

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

      3. unsub-neg N/A

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

      4. mul-1-neg N/A

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

      5. +-commutative N/A

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

      6. associate-+r+ N/A

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

      7. mul-1-neg N/A

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

      8. unsub-neg N/A

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

      9. *-commutative N/A

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

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

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

      11. lower-fma.f64 N/A

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

      12. lower--.f64 N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 64.4

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

   7. Simplified 64.4%

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

   8. Taylor expanded in t around inf

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

      1. lower-*.f64 46.1

         \[\leadsto \frac{\color{blue}{t \cdot x}}{t} \]

   10. Simplified 46.1%

       \[\leadsto \frac{\color{blue}{t \cdot x}}{t} \]
   11. Step-by-step derivation

       1. *-commutative N/A

          \[\leadsto \frac{\color{blue}{x \cdot t}}{t} \]

       2. associate-/l* N/A

          \[\leadsto \color{blue}{x \cdot \frac{t}{t}} \]

       3. *-inverses N/A

          \[\leadsto x \cdot \color{blue}{1} \]

       4. metadata-eval N/A

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

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

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

       6. *-commutative N/A

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

       7. neg-mul-1 N/A

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

       8. remove-double-neg 69.3

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

   12. Applied egg-rr 69.3%

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

   if -2.6000000000000002e-29 < t < 8.5000000000000002e90

   1. Initial program 97.3%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower-/.f64 47.6

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

   5. Simplified 47.6%

      \[\leadsto \color{blue}{z \cdot \frac{y}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+258}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{z \cdot \left(-x\right)}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.3e+258) (fma (/ z t) y x) (/ (* z (- x)) t)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.3e+258) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = (z * -x) / t;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.3e+258)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(Float64(z * Float64(-x)) / t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.3e+258], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(z * (-x)), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000005e258

   1. Initial program 90.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 77.3

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

   5. Simplified 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 80.7

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

   7. Applied egg-rr 80.7%

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

   if 1.30000000000000005e258 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      4. lower--.f64 68.0

         \[\leadsto z \cdot \frac{\color{blue}{y - x}}{t} \]

   5. Simplified 68.0%

      \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto \color{blue}{-1 \cdot \frac{x \cdot z}{t}} \]
   7. Step-by-step derivation

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. associate-*r* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. mul-1-neg N/A

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

      7. lower-neg.f64 68.3

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

   8. Simplified 68.3%

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

Alternative 7: 76.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+258}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \frac{x}{t}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x 1.3e+258) (fma (/ z t) y x) (- (* z (/ x t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= 1.3e+258) {
		tmp = fma((z / t), y, x);
	} else {
		tmp = -(z * (x / t));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= 1.3e+258)
		tmp = fma(Float64(z / t), y, x);
	else
		tmp = Float64(-Float64(z * Float64(x / t)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, 1.3e+258], N[(N[(z / t), $MachinePrecision] * y + x), $MachinePrecision], (-N[(z * N[(x / t), $MachinePrecision]), $MachinePrecision])]

Derivation

1. Split input into 2 regimes

2. if x < 1.30000000000000005e258

   1. Initial program 90.9%

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

   3. Taylor expanded in y around inf

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

      1. lower-*.f64 77.3

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

   5. Simplified 77.3%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. lift-*.f64 N/A

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

      6. associate-/l* N/A

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

      7. lift-/.f64 N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 80.7

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

   7. Applied egg-rr 80.7%

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

   if 1.30000000000000005e258 < x

   1. Initial program 93.4%

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

   3. Taylor expanded in z around inf

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

      1. div-sub N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      2. lower-*.f64 N/A

         \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

      3. lower-/.f64 N/A

         \[\leadsto z \cdot \color{blue}{\frac{y - x}{t}} \]

      4. lower--.f64 68.0

         \[\leadsto z \cdot \frac{\color{blue}{y - x}}{t} \]

   5. Simplified 68.0%

      \[\leadsto \color{blue}{z \cdot \frac{y - x}{t}} \]

   6. Taylor expanded in y around 0

      \[\leadsto z \cdot \frac{\color{blue}{-1 \cdot x}}{t} \]
   7. Step-by-step derivation

      1. mul-1-neg N/A

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

      2. lower-neg.f64 68.0

         \[\leadsto z \cdot \frac{\color{blue}{-x}}{t} \]

   8. Simplified 68.0%

      \[\leadsto z \cdot \frac{\color{blue}{-x}}{t} \]
3. Recombined 2 regimes into one program.

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq 1.3 \cdot 10^{+258}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;-z \cdot \frac{x}{t}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 77.4% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 75.5

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

5. Simplified 75.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-/l* N/A

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

   7. lift-/.f64 N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 78.4

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

7. Applied egg-rr 78.4%

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

Alternative 9: 73.9% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

3. Taylor expanded in y around inf

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

   1. lower-*.f64 75.5

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

5. Simplified 75.5%

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

   1. lift-*.f64 N/A

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

   2. lift-/.f64 N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. lift-*.f64 N/A

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

   6. associate-*l/ N/A

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

   7. lift-/.f64 N/A

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

   8. lower-fma.f64 75.5

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

7. Applied egg-rr 75.5%

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

Alternative 10: 38.8% accurate, 23.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 91.0%

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

   1. lift--.f64 N/A

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

   2. lift-*.f64 N/A

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

   3. lift-/.f64 N/A

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

   4. +-commutative N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lower-fma.f64 N/A

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

   10. lower-/.f64 98.0

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

4. Applied egg-rr 98.0%

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

5. Taylor expanded in t around 0

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

   1. lower-/.f64 N/A

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

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

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

   3. unsub-neg N/A

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

   4. mul-1-neg N/A

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

   5. +-commutative N/A

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

   6. associate-+r+ N/A

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

   7. mul-1-neg N/A

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

   8. unsub-neg N/A

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

   9. *-commutative N/A

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

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

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

   11. lower-fma.f64 N/A

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

   12. lower--.f64 N/A

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

   13. *-commutative N/A

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

   14. lower-*.f64 79.9

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

7. Simplified 79.9%

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

8. Taylor expanded in t around inf

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

   1. lower-*.f64 33.4

      \[\leadsto \frac{\color{blue}{t \cdot x}}{t} \]

10. Simplified 33.4%

    \[\leadsto \frac{\color{blue}{t \cdot x}}{t} \]
11. Step-by-step derivation

    1. *-commutative N/A

       \[\leadsto \frac{\color{blue}{x \cdot t}}{t} \]

    2. associate-/l* N/A

       \[\leadsto \color{blue}{x \cdot \frac{t}{t}} \]

    3. *-inverses N/A

       \[\leadsto x \cdot \color{blue}{1} \]

    4. metadata-eval N/A

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

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

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

    6. *-commutative N/A

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

    7. neg-mul-1 N/A

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

    8. remove-double-neg 44.8

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

12. Applied egg-rr 44.8%

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

Developer Target 1: 98.1% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x < -9.025511195533005 \cdot 10^{-135}:\\ \;\;\;\;x - \frac{z}{t} \cdot \left(x - y\right)\\ \mathbf{elif}\;x < 4.275032163700715 \cdot 10^{-250}:\\ \;\;\;\;x + \frac{y - x}{t} \cdot z\\ \mathbf{else}:\\ \;\;\;\;x + \frac{y - x}{\frac{t}{z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< x -9.025511195533005e-135)
   (- x (* (/ z t) (- x y)))
   (if (< x 4.275032163700715e-250)
     (+ x (* (/ (- y x) t) z))
     (+ x (/ (- y x) (/ t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x < (-9.025511195533005d-135)) then
        tmp = x - ((z / t) * (x - y))
    else if (x < 4.275032163700715d-250) then
        tmp = x + (((y - x) / t) * z)
    else
        tmp = x + ((y - x) / (t / z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x < -9.025511195533005e-135) {
		tmp = x - ((z / t) * (x - y));
	} else if (x < 4.275032163700715e-250) {
		tmp = x + (((y - x) / t) * z);
	} else {
		tmp = x + ((y - x) / (t / z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x < -9.025511195533005e-135:
		tmp = x - ((z / t) * (x - y))
	elif x < 4.275032163700715e-250:
		tmp = x + (((y - x) / t) * z)
	else:
		tmp = x + ((y - x) / (t / z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x < -9.025511195533005e-135)
		tmp = Float64(x - Float64(Float64(z / t) * Float64(x - y)));
	elseif (x < 4.275032163700715e-250)
		tmp = Float64(x + Float64(Float64(Float64(y - x) / t) * z));
	else
		tmp = Float64(x + Float64(Float64(y - x) / Float64(t / z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x < -9.025511195533005e-135)
		tmp = x - ((z / t) * (x - y));
	elseif (x < 4.275032163700715e-250)
		tmp = x + (((y - x) / t) * z);
	else
		tmp = x + ((y - x) / (t / z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[x, -9.025511195533005e-135], N[(x - N[(N[(z / t), $MachinePrecision] * N[(x - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[Less[x, 4.275032163700715e-250], N[(x + N[(N[(N[(y - x), $MachinePrecision] / t), $MachinePrecision] * z), $MachinePrecision]), $MachinePrecision], N[(x + N[(N[(y - x), $MachinePrecision] / N[(t / z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Reproduce

herbie shell --seed 2024212 
(FPCore (x y z t)
  :name "Numeric.Histogram:binBounds from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (if (< x -1805102239106601/200000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (- x (* (/ z t) (- x y))) (if (< x 855006432740143/2000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000000) (+ x (* (/ (- y x) t) z)) (+ x (/ (- y x) (/ t z))))))

  (+ x (/ (* (- y x) z) t)))