Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 80.1% → 94.7%

Time: 10.4s

Alternatives: 15

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-274}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a - z} - \frac{z}{a - z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_1 -4e-274)
     (fma (- t x) (/ (- y z) (- a z)) x)
     (if (<= t_1 0.0)
       (fma (- x t) (/ (- y a) z) t)
       (fma (- t x) (- (/ y (- a z)) (/ z (- a z))) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_1 <= -4e-274) {
		tmp = fma((t - x), ((y - z) / (a - z)), x);
	} else if (t_1 <= 0.0) {
		tmp = fma((x - t), ((y - a) / z), t);
	} else {
		tmp = fma((t - x), ((y / (a - z)) - (z / (a - z))), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_1 <= -4e-274)
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x);
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	else
		tmp = fma(Float64(t - x), Float64(Float64(y / Float64(a - z)) - Float64(z / Float64(a - z))), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e-274], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], N[(N[(t - x), $MachinePrecision] * N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] - N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -3.99999999999999986e-274

   1. Initial program 92.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 98.5

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

   4. Applied rewrites 98.5%

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

   if -3.99999999999999986e-274 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 6.4

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

   4. Applied rewrites 6.4%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - t, \frac{\color{blue}{y - a}}{z}, t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 97.6%

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

   if 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 88.3%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 92.3

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

   4. Applied rewrites 92.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)} \]
   5. Step-by-step derivation

      1. lift-/.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-sub N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower-/.f64 92.3

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

   6. Applied rewrites 92.3%

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

Alternative 2: 94.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{-274} \lor \neg \left(t\_1 \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (or (<= t_1 -4e-274) (not (<= t_1 0.0)))
     (fma (- t x) (/ (- y z) (- a z)) x)
     (fma (- x t) (/ (- y a) z) t))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if ((t_1 <= -4e-274) || !(t_1 <= 0.0)) {
		tmp = fma((t - x), ((y - z) / (a - z)), x);
	} else {
		tmp = fma((x - t), ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if ((t_1 <= -4e-274) || !(t_1 <= 0.0))
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / Float64(a - z)), x);
	else
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[Or[LessEqual[t$95$1, -4e-274], N[Not[LessEqual[t$95$1, 0.0]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -3.99999999999999986e-274 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 90.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 95.0

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

   4. Applied rewrites 95.0%

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

   if -3.99999999999999986e-274 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 6.4

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

   4. Applied rewrites 6.4%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 97.6%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - t, \frac{\color{blue}{y - a}}{z}, t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 97.6%

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

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq -4 \cdot 10^{-274} \lor \neg \left(x + \left(y - z\right) \cdot \frac{t - x}{a - z} \leq 0\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 60.3% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{if}\;a \leq -0.041:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-192}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- t x) a) y x)))
   (if (<= a -0.041)
     t_1
     (if (<= a 2.2e-192)
       (fma (- t) (/ y z) t)
       (if (<= a 1.25e+35) (fma (/ x z) y t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((t - x) / a), y, x);
	double tmp;
	if (a <= -0.041) {
		tmp = t_1;
	} else if (a <= 2.2e-192) {
		tmp = fma(-t, (y / z), t);
	} else if (a <= 1.25e+35) {
		tmp = fma((x / z), y, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(t - x) / a), y, x)
	tmp = 0.0
	if (a <= -0.041)
		tmp = t_1;
	elseif (a <= 2.2e-192)
		tmp = fma(Float64(-t), Float64(y / z), t);
	elseif (a <= 1.25e+35)
		tmp = fma(Float64(x / z), y, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[a, -0.041], t$95$1, If[LessEqual[a, 2.2e-192], N[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[a, 1.25e+35], N[(N[(x / z), $MachinePrecision] * y + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -0.0410000000000000017 or 1.25000000000000005e35 < a

   1. Initial program 92.4%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 75.2

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

   5. Applied rewrites 75.2%

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

   if -0.0410000000000000017 < a < 2.20000000000000006e-192

   1. Initial program 71.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 77.9

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

   4. Applied rewrites 77.9%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 82.3%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 75.4%

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

   11. Taylor expanded in x around 0

       \[\leadsto t + -1 \cdot \color{blue}{\frac{t \cdot y}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 64.6%

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

   if 2.20000000000000006e-192 < a < 1.25000000000000005e35

   1. Initial program 67.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 73.6

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

   4. Applied rewrites 73.6%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 78.1%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 67.3%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{x}{z}, y, t\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 60.1%

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

4. Final simplification 68.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.041:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-192}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{elif}\;a \leq 1.25 \cdot 10^{+35}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 48.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{z}{a - z}, x\right)\\ \mathbf{if}\;a \leq -2.9 \cdot 10^{+140}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq -0.45:\\ \;\;\;\;t \cdot \frac{y - z}{a}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma x (/ z (- a z)) x)))
   (if (<= a -2.9e+140)
     t_1
     (if (<= a -0.45)
       (* t (/ (- y z) a))
       (if (<= a 1.65e+66) (fma (- t) (/ y z) t) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(x, (z / (a - z)), x);
	double tmp;
	if (a <= -2.9e+140) {
		tmp = t_1;
	} else if (a <= -0.45) {
		tmp = t * ((y - z) / a);
	} else if (a <= 1.65e+66) {
		tmp = fma(-t, (y / z), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(x, Float64(z / Float64(a - z)), x)
	tmp = 0.0
	if (a <= -2.9e+140)
		tmp = t_1;
	elseif (a <= -0.45)
		tmp = Float64(t * Float64(Float64(y - z) / a));
	elseif (a <= 1.65e+66)
		tmp = fma(Float64(-t), Float64(y / z), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.9e+140], t$95$1, If[LessEqual[a, -0.45], N[(t * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.65e+66], N[((-t) * N[(y / z), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if a < -2.8999999999999999e140 or 1.6500000000000001e66 < a

   1. Initial program 94.4%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. associate-/l* N/A

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

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

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

      5. lower-fma.f64 N/A

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

      6. lower-neg.f64 N/A

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

      7. lower-/.f64 N/A

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

      8. lower--.f64 N/A

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

      9. lower--.f64 69.6

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

   5. Applied rewrites 69.6%

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

   6. Taylor expanded in t around 0

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

   8. Applied rewrites 57.8%

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

   if -2.8999999999999999e140 < a < -0.450000000000000011

   1. Initial program 91.4%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 66.4

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

   5. Applied rewrites 66.4%

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

   6. Taylor expanded in a around inf

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

   8. Applied rewrites 48.3%

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

   if -0.450000000000000011 < a < 1.6500000000000001e66

   1. Initial program 70.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 71.9%

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

   11. Taylor expanded in x around 0

       \[\leadsto t + -1 \cdot \color{blue}{\frac{t \cdot y}{z}} \]
   12. Step-by-step derivation

   13. Applied rewrites 59.9%

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

4. Final simplification 58.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -2.9 \cdot 10^{+140}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a - z}, x\right)\\ \mathbf{elif}\;a \leq -0.45:\\ \;\;\;\;t \cdot \frac{y - z}{a}\\ \mathbf{elif}\;a \leq 1.65 \cdot 10^{+66}:\\ \;\;\;\;\mathsf{fma}\left(-t, \frac{y}{z}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a - z}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 76.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.52 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.52) (not (<= a 1.06e+66)))
   (fma (- t x) (/ (- y z) a) x)
   (fma (- x t) (/ (- y a) z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.52) || !(a <= 1.06e+66)) {
		tmp = fma((t - x), ((y - z) / a), x);
	} else {
		tmp = fma((x - t), ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.52) || !(a <= 1.06e+66))
		tmp = fma(Float64(t - x), Float64(Float64(y - z) / a), x);
	else
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.52], N[Not[LessEqual[a, 1.06e+66]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * N[(N[(y - z), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.52000000000000002 or 1.06000000000000004e66 < a

   1. Initial program 93.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in a around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 86.6

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

   7. Applied rewrites 86.6%

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

   if -0.52000000000000002 < a < 1.06000000000000004e66

   1. Initial program 70.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - t, \frac{\color{blue}{y - a}}{z}, t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.8%

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

4. Final simplification 83.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.52 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y - z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 75.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.52 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.52) (not (<= a 1.06e+66)))
   (fma (- y z) (/ (- t x) a) x)
   (fma (- x t) (/ (- y a) z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.52) || !(a <= 1.06e+66)) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = fma((x - t), ((y - a) / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.52) || !(a <= 1.06e+66))
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = fma(Float64(x - t), Float64(Float64(y - a) / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.52], N[Not[LessEqual[a, 1.06e+66]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.52000000000000002 or 1.06000000000000004e66 < a

   1. Initial program 93.7%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -0.52000000000000002 < a < 1.06000000000000004e66

   1. Initial program 70.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - t, \frac{\color{blue}{y - a}}{z}, t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.8%

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

4. Final simplification 82.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.52 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y - a}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 73.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.37 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.37) (not (<= a 1.06e+66)))
   (fma (- y z) (/ (- t x) a) x)
   (fma (- x t) (/ y z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.37) || !(a <= 1.06e+66)) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = fma((x - t), (y / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.37) || !(a <= 1.06e+66))
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = fma(Float64(x - t), Float64(y / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.37], N[Not[LessEqual[a, 1.06e+66]], $MachinePrecision]], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.37 or 1.06000000000000004e66 < a

   1. Initial program 93.7%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

      4. lower-fma.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 N/A

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

      7. lower--.f64 85.7

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

   5. Applied rewrites 85.7%

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

   if -0.37 < a < 1.06000000000000004e66

   1. Initial program 70.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - t, \frac{\color{blue}{y - a}}{z}, t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.8%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.3%

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

4. Final simplification 80.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.37 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 70.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.37 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.37) (not (<= a 1.06e+66)))
   (fma (- t x) (/ y a) x)
   (fma (- x t) (/ y z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.37) || !(a <= 1.06e+66)) {
		tmp = fma((t - x), (y / a), x);
	} else {
		tmp = fma((x - t), (y / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.37) || !(a <= 1.06e+66))
		tmp = fma(Float64(t - x), Float64(y / a), x);
	else
		tmp = fma(Float64(x - t), Float64(y / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.37], N[Not[LessEqual[a, 1.06e+66]], $MachinePrecision]], N[(N[(t - x), $MachinePrecision] * N[(y / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.37 or 1.06000000000000004e66 < a

   1. Initial program 93.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 96.0

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

   4. Applied rewrites 96.0%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma}\left(t - x, \color{blue}{\frac{y}{a}}, x\right) \]
   6. Step-by-step derivation

      1. lower-/.f64 77.4

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

   7. Applied rewrites 77.4%

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

   if -0.37 < a < 1.06000000000000004e66

   1. Initial program 70.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - t, \frac{\color{blue}{y - a}}{z}, t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.8%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.3%

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

4. Final simplification 76.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.37 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{y}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.37 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.37) (not (<= a 1.06e+66)))
   (fma (/ (- t x) a) y x)
   (fma (- x t) (/ y z) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.37) || !(a <= 1.06e+66)) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = fma((x - t), (y / z), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.37) || !(a <= 1.06e+66))
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = fma(Float64(x - t), Float64(y / z), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.37], N[Not[LessEqual[a, 1.06e+66]], $MachinePrecision]], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(y / z), $MachinePrecision] + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.37 or 1.06000000000000004e66 < a

   1. Initial program 93.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if -0.37 < a < 1.06000000000000004e66

   1. Initial program 70.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in x around 0

      \[\leadsto \mathsf{fma}\left(x - t, \frac{\color{blue}{y - a}}{z}, t\right) \]
   9. Step-by-step derivation

   10. Applied rewrites 80.8%

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

   11. Taylor expanded in y around inf

       \[\leadsto \mathsf{fma}\left(x - t, \frac{y}{\color{blue}{z}}, t\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 76.3%

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

4. Final simplification 76.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.37 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{y}{z}, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -0.37 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= a -0.37) (not (<= a 1.06e+66)))
   (fma (/ (- t x) a) y x)
   (fma (/ (- x t) z) y t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((a <= -0.37) || !(a <= 1.06e+66)) {
		tmp = fma(((t - x) / a), y, x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((a <= -0.37) || !(a <= 1.06e+66))
		tmp = fma(Float64(Float64(t - x) / a), y, x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[a, -0.37], N[Not[LessEqual[a, 1.06e+66]], $MachinePrecision]], N[(N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if a < -0.37 or 1.06000000000000004e66 < a

   1. Initial program 93.7%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-commutative N/A

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

      4. lower-fma.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 77.1

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

   5. Applied rewrites 77.1%

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

   if -0.37 < a < 1.06000000000000004e66

   1. Initial program 70.7%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 77.1

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

   4. Applied rewrites 77.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 80.8%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 71.9%

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

4. Final simplification 74.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -0.37 \lor \neg \left(a \leq 1.06 \cdot 10^{+66}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 45.2% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+26} \lor \neg \left(z \leq 2.1 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -9.5e+26) (not (<= z 2.1e-30)))
   (fma (/ x z) y t)
   (* t (/ y (- a z)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -9.5e+26) || !(z <= 2.1e-30)) {
		tmp = fma((x / z), y, t);
	} else {
		tmp = t * (y / (a - z));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -9.5e+26) || !(z <= 2.1e-30))
		tmp = fma(Float64(x / z), y, t);
	else
		tmp = Float64(t * Float64(y / Float64(a - z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -9.5e+26], N[Not[LessEqual[z, 2.1e-30]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * y + t), $MachinePrecision], N[(t * N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -9.50000000000000054e26 or 2.1000000000000002e-30 < z

   1. Initial program 70.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 75.1

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

   4. Applied rewrites 75.1%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 74.2%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 67.1%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{x}{z}, y, t\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 60.0%

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

   if -9.50000000000000054e26 < z < 2.1000000000000002e-30

   1. Initial program 89.5%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 45.2

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

   5. Applied rewrites 45.2%

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

   6. Taylor expanded in y around inf

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

   8. Applied rewrites 43.9%

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

4. Final simplification 51.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+26} \lor \neg \left(z \leq 2.1 \cdot 10^{-30}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 43.6% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-17} \lor \neg \left(z \leq 2.1 \cdot 10^{-44}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -6e-17) (not (<= z 2.1e-44))) (fma (/ x z) y t) (* t (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -6e-17) || !(z <= 2.1e-44)) {
		tmp = fma((x / z), y, t);
	} else {
		tmp = t * (y / a);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -6e-17) || !(z <= 2.1e-44))
		tmp = fma(Float64(x / z), y, t);
	else
		tmp = Float64(t * Float64(y / a));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -6e-17], N[Not[LessEqual[z, 2.1e-44]], $MachinePrecision]], N[(N[(x / z), $MachinePrecision] * y + t), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6.00000000000000012e-17 or 2.10000000000000001e-44 < z

   1. Initial program 73.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. associate-*l/ N/A

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

      7. associate-/l* N/A

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

      8. lower-fma.f64 N/A

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

      9. lower-/.f64 77.7

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

   4. Applied rewrites 77.7%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

      2. associate-*r/ N/A

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

      3. associate-*r* N/A

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

      4. mul-1-neg N/A

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

      5. associate-*r/ N/A

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

      6. div-sub N/A

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

      7. mul-1-neg N/A

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

      8. associate-*r* N/A

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

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

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

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

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

      11. associate-*r/ N/A

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

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

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

   7. Applied rewrites 72.4%

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

   8. Taylor expanded in a around 0

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

   10. Applied rewrites 65.8%

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

   11. Taylor expanded in x around inf

       \[\leadsto \mathsf{fma}\left(\frac{x}{z}, y, t\right) \]
   12. Step-by-step derivation

   13. Applied rewrites 54.8%

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

   if -6.00000000000000012e-17 < z < 2.10000000000000001e-44

   1. Initial program 88.6%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 42.0

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

   5. Applied rewrites 42.0%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 33.5%

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

4. Final simplification 45.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6 \cdot 10^{-17} \lor \neg \left(z \leq 2.1 \cdot 10^{-44}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 30.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+42} \lor \neg \left(z \leq 2.8 \cdot 10^{-43}\right):\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.1e+42) (not (<= z 2.8e-43))) (+ x (- t x)) (* t (/ y a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.1e+42) || !(z <= 2.8e-43)) {
		tmp = x + (t - x);
	} else {
		tmp = t * (y / a);
	}
	return tmp;
}

Fortran

real(8) function code(x, y, z, t, a)
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if ((z <= (-1.1d+42)) .or. (.not. (z <= 2.8d-43))) then
        tmp = x + (t - x)
    else
        tmp = t * (y / a)
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.1e+42) || !(z <= 2.8e-43)) {
		tmp = x + (t - x);
	} else {
		tmp = t * (y / a);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1.1e+42) or not (z <= 2.8e-43):
		tmp = x + (t - x)
	else:
		tmp = t * (y / a)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.1e+42) || !(z <= 2.8e-43))
		tmp = Float64(x + Float64(t - x));
	else
		tmp = Float64(t * Float64(y / a));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1.1e+42) || ~((z <= 2.8e-43)))
		tmp = x + (t - x);
	else
		tmp = t * (y / a);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.1e+42], N[Not[LessEqual[z, 2.8e-43]], $MachinePrecision]], N[(x + N[(t - x), $MachinePrecision]), $MachinePrecision], N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.1000000000000001e42 or 2.7999999999999998e-43 < z

   1. Initial program 71.0%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 35.6

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

   5. Applied rewrites 35.6%

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

   if -1.1000000000000001e42 < z < 2.7999999999999998e-43

   1. Initial program 88.7%

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

   3. Taylor expanded in x around 0

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 44.7

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

   5. Applied rewrites 44.7%

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

   6. Taylor expanded in z around 0

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

   8. Applied rewrites 31.2%

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

4. Final simplification 33.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.1 \cdot 10^{+42} \lor \neg \left(z \leq 2.8 \cdot 10^{-43}\right):\\ \;\;\;\;x + \left(t - x\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 18.9% accurate, 4.1× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 79.8%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 19.7

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

5. Applied rewrites 19.7%

   \[\leadsto x + \color{blue}{\left(t - x\right)} \]
6. Add Preprocessing

Alternative 15: 2.8% accurate, 4.8× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := N[(x + (-x)), $MachinePrecision]

Derivation

1. Initial program 79.8%

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

3. Taylor expanded in z around inf

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

   1. lower--.f64 19.7

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

5. Applied rewrites 19.7%

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

6. Taylor expanded in x around inf

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

8. Applied rewrites 2.8%

   \[\leadsto x + \left(-x\right) \]
9. Add Preprocessing

Reproduce

herbie shell --seed 2024337 
(FPCore (x y z t a)
  :name "Numeric.Signal:interpolate   from hsignal-0.2.7.1"
  :precision binary64
  (+ x (* (- y z) (/ (- t x) (- a z)))))