Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3

Percentage Accurate: 90.1% → 98.5%

Time: 7.3s

Alternatives: 8

Speedup: 1.3×

• 24.5% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 90.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.5% accurate, 0.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{if}\;t_1 \leq -\infty \lor \neg \left(t_1 \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(y, -z, y \cdot z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (* (- (* x y) (* y z)) t)))
   (if (or (<= t_1 (- INFINITY)) (not (<= t_1 5e+303)))
     (* y (* t (- x z)))
     (fma (* y (- x z)) t (* t (fma y (- z) (* y z)))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((x * y) - (y * z)) * t;
	double tmp;
	if ((t_1 <= -((double) INFINITY)) || !(t_1 <= 5e+303)) {
		tmp = y * (t * (x - z));
	} else {
		tmp = fma((y * (x - z)), t, (t * fma(y, -z, (y * z))));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(x * y) - Float64(y * z)) * t)
	tmp = 0.0
	if ((t_1 <= Float64(-Inf)) || !(t_1 <= 5e+303))
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = fma(Float64(y * Float64(x - z)), t, Float64(t * fma(y, Float64(-z), Float64(y * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]}, If[Or[LessEqual[t$95$1, (-Infinity)], N[Not[LessEqual[t$95$1, 5e+303]], $MachinePrecision]], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t + N[(t * N[(y * (-z) + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < -inf.0 or 4.9999999999999997e303 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t)

   1. Initial program 73.0%

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

      1. distribute-rgt-out-- 78.6%

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

      2. associate-*l* 100.0%

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

   3. Simplified 100.0%

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

   if -inf.0 < (*.f64 (-.f64 (*.f64 x y) (*.f64 z y)) t) < 4.9999999999999997e303

   1. Initial program 98.0%

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

      1. distribute-rgt-out-- 98.0%

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

      2. associate-*l* 85.2%

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

   3. Simplified 85.2%

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

      1. associate-*r* 98.0%

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

      2. distribute-rgt-out-- 98.0%

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

      3. *-commutative 98.0%

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

      4. prod-diff 98.0%

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

      5. *-commutative 98.0%

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

      6. fma-neg 98.0%

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

      7. distribute-rgt-in 98.1%

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

      8. fma-def 98.2%

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

      9. distribute-rgt-out-- 98.2%

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

      10. *-commutative 98.2%

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

      11. fma-udef 98.0%

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

      12. distribute-lft-neg-in 98.0%

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

      13. distribute-rgt-neg-in 98.0%

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

      14. fma-def 98.2%

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

      15. *-commutative 98.2%

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

   5. Applied egg-rr 98.2%

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

4. Final simplification 98.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(x \cdot y - y \cdot z\right) \cdot t \leq -\infty \lor \neg \left(\left(x \cdot y - y \cdot z\right) \cdot t \leq 5 \cdot 10^{+303}\right):\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y \cdot \left(x - z\right), t, t \cdot \mathsf{fma}\left(y, -z, y \cdot z\right)\right)\\ \end{array} \]

Alternative 2: 98.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x \cdot y - y \cdot z\\ \mathbf{if}\;t_1 \leq -1 \cdot 10^{+298}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;t_1 \leq 10^{+174}:\\ \;\;\;\;t_1 \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{\frac{1}{x - z}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (* x y) (* y z))))
   (if (<= t_1 -1e+298)
     (* y (* t (- x z)))
     (if (<= t_1 1e+174) (* t_1 t) (* y (/ t (/ 1.0 (- x z))))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x * y) - (y * z);
	double tmp;
	if (t_1 <= -1e+298) {
		tmp = y * (t * (x - z));
	} else if (t_1 <= 1e+174) {
		tmp = t_1 * t;
	} else {
		tmp = y * (t / (1.0 / (x - z)));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: t_1
    real(8) :: tmp
    t_1 = (x * y) - (y * z)
    if (t_1 <= (-1d+298)) then
        tmp = y * (t * (x - z))
    else if (t_1 <= 1d+174) then
        tmp = t_1 * t
    else
        tmp = y * (t / (1.0d0 / (x - z)))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x * y) - (y * z);
	double tmp;
	if (t_1 <= -1e+298) {
		tmp = y * (t * (x - z));
	} else if (t_1 <= 1e+174) {
		tmp = t_1 * t;
	} else {
		tmp = y * (t / (1.0 / (x - z)));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x * y) - (y * z)
	tmp = 0
	if t_1 <= -1e+298:
		tmp = y * (t * (x - z))
	elif t_1 <= 1e+174:
		tmp = t_1 * t
	else:
		tmp = y * (t / (1.0 / (x - z)))
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x * y) - Float64(y * z))
	tmp = 0.0
	if (t_1 <= -1e+298)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	elseif (t_1 <= 1e+174)
		tmp = Float64(t_1 * t);
	else
		tmp = Float64(y * Float64(t / Float64(1.0 / Float64(x - z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x * y) - (y * z);
	tmp = 0.0;
	if (t_1 <= -1e+298)
		tmp = y * (t * (x - z));
	elseif (t_1 <= 1e+174)
		tmp = t_1 * t;
	else
		tmp = y * (t / (1.0 / (x - z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x * y), $MachinePrecision] - N[(y * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -1e+298], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+174], N[(t$95$1 * t), $MachinePrecision], N[(y * N[(t / N[(1.0 / N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (-.f64 (*.f64 x y) (*.f64 z y)) < -9.9999999999999996e297

   1. Initial program 61.5%

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

      1. distribute-rgt-out-- 61.5%

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

      2. associate-*l* 99.9%

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

   3. Simplified 99.9%

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

   if -9.9999999999999996e297 < (-.f64 (*.f64 x y) (*.f64 z y)) < 1.00000000000000007e174

   1. Initial program 98.1%

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

   if 1.00000000000000007e174 < (-.f64 (*.f64 x y) (*.f64 z y))

   1. Initial program 76.4%

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

      1. distribute-rgt-out-- 85.5%

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

      2. associate-*l* 99.8%

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

   3. Simplified 99.8%

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

      1. *-commutative 99.8%

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

      2. flip-- 58.1%

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

      3. associate-*r/ 58.1%

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

   5. Applied egg-rr 58.1%

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

      1. associate-/l* 58.1%

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

      2. difference-of-squares 68.9%

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

      3. associate-/r* 99.8%

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

      4. *-inverses 99.8%

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

   7. Simplified 99.8%

      \[\leadsto y \cdot \color{blue}{\frac{t}{\frac{1}{x - z}}} \]
3. Recombined 3 regimes into one program.

4. Final simplification 98.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \cdot y - y \cdot z \leq -1 \cdot 10^{+298}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{elif}\;x \cdot y - y \cdot z \leq 10^{+174}:\\ \;\;\;\;\left(x \cdot y - y \cdot z\right) \cdot t\\ \mathbf{else}:\\ \;\;\;\;y \cdot \frac{t}{\frac{1}{x - z}}\\ \end{array} \]

Alternative 3: 72.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 10^{-18}:\\ \;\;\;\;-y \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -2.45e-37)
   (* x (* y t))
   (if (<= x 1e-18) (- (* y (* z t))) (* y (* x t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.45e-37) {
		tmp = x * (y * t);
	} else if (x <= 1e-18) {
		tmp = -(y * (z * t));
	} else {
		tmp = y * (x * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-2.45d-37)) then
        tmp = x * (y * t)
    else if (x <= 1d-18) then
        tmp = -(y * (z * t))
    else
        tmp = y * (x * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -2.45e-37) {
		tmp = x * (y * t);
	} else if (x <= 1e-18) {
		tmp = -(y * (z * t));
	} else {
		tmp = y * (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -2.45e-37:
		tmp = x * (y * t)
	elif x <= 1e-18:
		tmp = -(y * (z * t))
	else:
		tmp = y * (x * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -2.45e-37)
		tmp = Float64(x * Float64(y * t));
	elseif (x <= 1e-18)
		tmp = Float64(-Float64(y * Float64(z * t)));
	else
		tmp = Float64(y * Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -2.45e-37)
		tmp = x * (y * t);
	elseif (x <= 1e-18)
		tmp = -(y * (z * t));
	else
		tmp = y * (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -2.45e-37], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 1e-18], (-N[(y * N[(z * t), $MachinePrecision]), $MachinePrecision]), N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -2.45000000000000009e-37

   1. Initial program 88.1%

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

      1. distribute-rgt-out-- 92.1%

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

      2. associate-*l* 89.5%

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

   3. Simplified 89.5%

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

      1. associate-*r* 92.1%

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

      2. distribute-rgt-out-- 88.1%

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

      3. *-commutative 88.1%

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

      4. prod-diff 88.0%

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

      5. *-commutative 88.0%

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

      6. fma-neg 88.0%

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

      7. distribute-rgt-in 85.3%

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

      8. fma-def 85.4%

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

      9. distribute-rgt-out-- 86.7%

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

      10. *-commutative 86.7%

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

      11. fma-udef 88.0%

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

      12. distribute-lft-neg-in 88.0%

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

      13. distribute-rgt-neg-in 88.0%

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

      14. fma-def 86.7%

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

      15. *-commutative 86.7%

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

   5. Applied egg-rr 86.7%

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

   6. Taylor expanded in z around 0 81.3%

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

   7. Taylor expanded in z around 0 72.0%

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

      1. associate-*r* 74.6%

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

   9. Simplified 74.6%

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

   if -2.45000000000000009e-37 < x < 1.0000000000000001e-18

   1. Initial program 95.4%

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

      1. distribute-rgt-out-- 95.4%

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

      2. associate-*l* 90.6%

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

   3. Simplified 90.6%

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

   4. Taylor expanded in x around 0 75.5%

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

      1. associate-*r* 75.5%

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

      2. neg-mul-1 75.5%

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

   6. Simplified 75.5%

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

   if 1.0000000000000001e-18 < x

   1. Initial program 86.3%

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

      1. distribute-rgt-out-- 87.9%

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

      2. associate-*l* 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in x around inf 74.5%

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

4. Final simplification 75.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -2.45 \cdot 10^{-37}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 10^{-18}:\\ \;\;\;\;-y \cdot \left(z \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 4: 73.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-15}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= x -3.1e-36)
   (* x (* y t))
   (if (<= x 8.2e-15) (* t (* y (- z))) (* y (* x t)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.1e-36) {
		tmp = x * (y * t);
	} else if (x <= 8.2e-15) {
		tmp = t * (y * -z);
	} else {
		tmp = y * (x * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (x <= (-3.1d-36)) then
        tmp = x * (y * t)
    else if (x <= 8.2d-15) then
        tmp = t * (y * -z)
    else
        tmp = y * (x * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (x <= -3.1e-36) {
		tmp = x * (y * t);
	} else if (x <= 8.2e-15) {
		tmp = t * (y * -z);
	} else {
		tmp = y * (x * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if x <= -3.1e-36:
		tmp = x * (y * t)
	elif x <= 8.2e-15:
		tmp = t * (y * -z)
	else:
		tmp = y * (x * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (x <= -3.1e-36)
		tmp = Float64(x * Float64(y * t));
	elseif (x <= 8.2e-15)
		tmp = Float64(t * Float64(y * Float64(-z)));
	else
		tmp = Float64(y * Float64(x * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (x <= -3.1e-36)
		tmp = x * (y * t);
	elseif (x <= 8.2e-15)
		tmp = t * (y * -z);
	else
		tmp = y * (x * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[x, -3.1e-36], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision], If[LessEqual[x, 8.2e-15], N[(t * N[(y * (-z)), $MachinePrecision]), $MachinePrecision], N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if x < -3.0999999999999999e-36

   1. Initial program 88.1%

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

      1. distribute-rgt-out-- 92.1%

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

      2. associate-*l* 89.5%

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

   3. Simplified 89.5%

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

      1. associate-*r* 92.1%

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

      2. distribute-rgt-out-- 88.1%

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

      3. *-commutative 88.1%

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

      4. prod-diff 88.0%

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

      5. *-commutative 88.0%

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

      6. fma-neg 88.0%

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

      7. distribute-rgt-in 85.3%

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

      8. fma-def 85.4%

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

      9. distribute-rgt-out-- 86.7%

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

      10. *-commutative 86.7%

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

      11. fma-udef 88.0%

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

      12. distribute-lft-neg-in 88.0%

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

      13. distribute-rgt-neg-in 88.0%

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

      14. fma-def 86.7%

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

      15. *-commutative 86.7%

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

   5. Applied egg-rr 86.7%

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

   6. Taylor expanded in z around 0 81.3%

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

   7. Taylor expanded in z around 0 72.0%

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

      1. associate-*r* 74.6%

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

   9. Simplified 74.6%

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

   if -3.0999999999999999e-36 < x < 8.20000000000000072e-15

   1. Initial program 95.4%

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

      1. distribute-rgt-out-- 95.4%

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

   3. Simplified 95.4%

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

   4. Taylor expanded in x around 0 79.7%

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

      1. neg-mul-1 79.7%

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

      2. distribute-lft-neg-in 79.7%

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

   6. Simplified 79.7%

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

   if 8.20000000000000072e-15 < x

   1. Initial program 86.3%

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

      1. distribute-rgt-out-- 87.9%

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

      2. associate-*l* 87.8%

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

   3. Simplified 87.8%

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

   4. Taylor expanded in x around inf 74.5%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x \leq -3.1 \cdot 10^{-36}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \mathbf{elif}\;x \leq 8.2 \cdot 10^{-15}:\\ \;\;\;\;t \cdot \left(y \cdot \left(-z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \end{array} \]

Alternative 5: 94.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 700000000000:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 700000000000.0) (* y (* t (- x z))) (* t (* y (- x z)))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 700000000000.0) {
		tmp = y * (t * (x - z));
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 700000000000.0d0) then
        tmp = y * (t * (x - z))
    else
        tmp = t * (y * (x - z))
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 700000000000.0) {
		tmp = y * (t * (x - z));
	} else {
		tmp = t * (y * (x - z));
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 700000000000.0:
		tmp = y * (t * (x - z))
	else:
		tmp = t * (y * (x - z))
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 700000000000.0)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(t * Float64(y * Float64(x - z)));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 700000000000.0)
		tmp = y * (t * (x - z));
	else
		tmp = t * (y * (x - z));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 700000000000.0], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t * N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 7e11

   1. Initial program 87.7%

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

      1. distribute-rgt-out-- 90.0%

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

      2. associate-*l* 90.5%

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

   3. Simplified 90.5%

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

   if 7e11 < t

   1. Initial program 98.5%

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

      1. distribute-rgt-out-- 98.5%

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

   3. Simplified 98.5%

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

4. Final simplification 92.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 700000000000:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \left(y \cdot \left(x - z\right)\right)\\ \end{array} \]

Alternative 6: 55.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-72}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t 1.65e-72) (* y (* x t)) (* x (* y t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.65e-72) {
		tmp = y * (x * t);
	} else {
		tmp = x * (y * t);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t <= 1.65d-72) then
        tmp = y * (x * t)
    else
        tmp = x * (y * t)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= 1.65e-72) {
		tmp = y * (x * t);
	} else {
		tmp = x * (y * t);
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= 1.65e-72:
		tmp = y * (x * t)
	else:
		tmp = x * (y * t)
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= 1.65e-72)
		tmp = Float64(y * Float64(x * t));
	else
		tmp = Float64(x * Float64(y * t));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= 1.65e-72)
		tmp = y * (x * t);
	else
		tmp = x * (y * t);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, 1.65e-72], N[(y * N[(x * t), $MachinePrecision]), $MachinePrecision], N[(x * N[(y * t), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < 1.65e-72

   1. Initial program 86.7%

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

      1. distribute-rgt-out-- 89.2%

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

      2. associate-*l* 89.4%

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

   3. Simplified 89.4%

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

   4. Taylor expanded in x around inf 55.6%

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

   if 1.65e-72 < t

   1. Initial program 97.8%

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

      1. distribute-rgt-out-- 97.8%

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

      2. associate-*l* 89.7%

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

   3. Simplified 89.7%

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

      1. associate-*r* 97.8%

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

      2. distribute-rgt-out-- 97.8%

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

      3. *-commutative 97.8%

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

      4. prod-diff 90.3%

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

      5. *-commutative 90.3%

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

      6. fma-neg 90.3%

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

      7. distribute-rgt-in 82.8%

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

      8. fma-def 83.0%

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

      9. distribute-rgt-out-- 83.0%

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

      10. *-commutative 83.0%

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

      11. fma-udef 90.3%

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

      12. distribute-lft-neg-in 90.3%

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

      13. distribute-rgt-neg-in 90.3%

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

      14. fma-def 83.0%

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

      15. *-commutative 83.0%

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

   5. Applied egg-rr 83.0%

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

   6. Taylor expanded in z around 0 88.7%

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

   7. Taylor expanded in z around 0 50.6%

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

      1. associate-*r* 55.7%

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

   9. Simplified 55.7%

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

4. Final simplification 55.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.65 \cdot 10^{-72}:\\ \;\;\;\;y \cdot \left(x \cdot t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \left(y \cdot t\right)\\ \end{array} \]

Alternative 7: 92.2% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. distribute-rgt-out-- 92.4%

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

   2. associate-*l* 89.5%

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

3. Simplified 89.5%

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

4. Final simplification 89.5%

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

Alternative 8: 53.7% accurate, 1.8× speedup

Math

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

FPCore

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

C

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 90.8%

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

   1. distribute-rgt-out-- 92.4%

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

   2. associate-*l* 89.5%

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

3. Simplified 89.5%

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

4. Taylor expanded in x around inf 53.8%

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

5. Final simplification 53.8%

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

Developer target: 96.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t < -9.231879582886777 \cdot 10^{-80}:\\ \;\;\;\;\left(y \cdot t\right) \cdot \left(x - z\right)\\ \mathbf{elif}\;t < 2.543067051564877 \cdot 10^{+83}:\\ \;\;\;\;y \cdot \left(t \cdot \left(x - z\right)\right)\\ \mathbf{else}:\\ \;\;\;\;\left(y \cdot \left(x - z\right)\right) \cdot t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (< t -9.231879582886777e-80)
   (* (* y t) (- x z))
   (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8) :: tmp
    if (t < (-9.231879582886777d-80)) then
        tmp = (y * t) * (x - z)
    else if (t < 2.543067051564877d+83) then
        tmp = y * (t * (x - z))
    else
        tmp = (y * (x - z)) * t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t < -9.231879582886777e-80) {
		tmp = (y * t) * (x - z);
	} else if (t < 2.543067051564877e+83) {
		tmp = y * (t * (x - z));
	} else {
		tmp = (y * (x - z)) * t;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t < -9.231879582886777e-80:
		tmp = (y * t) * (x - z)
	elif t < 2.543067051564877e+83:
		tmp = y * (t * (x - z))
	else:
		tmp = (y * (x - z)) * t
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t < -9.231879582886777e-80)
		tmp = Float64(Float64(y * t) * Float64(x - z));
	elseif (t < 2.543067051564877e+83)
		tmp = Float64(y * Float64(t * Float64(x - z)));
	else
		tmp = Float64(Float64(y * Float64(x - z)) * t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t < -9.231879582886777e-80)
		tmp = (y * t) * (x - z);
	elseif (t < 2.543067051564877e+83)
		tmp = y * (t * (x - z));
	else
		tmp = (y * (x - z)) * t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[Less[t, -9.231879582886777e-80], N[(N[(y * t), $MachinePrecision] * N[(x - z), $MachinePrecision]), $MachinePrecision], If[Less[t, 2.543067051564877e+83], N[(y * N[(t * N[(x - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(y * N[(x - z), $MachinePrecision]), $MachinePrecision] * t), $MachinePrecision]]]

Reproduce

herbie shell --seed 2023230 
(FPCore (x y z t)
  :name "Linear.Projection:inverseInfinitePerspective from linear-1.19.1.3"
  :precision binary64

  :herbie-target
  (if (< t -9.231879582886777e-80) (* (* y t) (- x z)) (if (< t 2.543067051564877e+83) (* y (* t (- x z))) (* (* y (- x z)) t)))

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