Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 79.5% → 94.4%

Time: 14.1s

Alternatives: 17

Speedup: 0.7×

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: 79.5% 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: 94.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{-243}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- y z) (- a z)) (- t x) x))
        (t_2 (+ x (* (- y z) (/ (- t x) (- a z))))))
   (if (<= t_2 -5e-243) t_1 (if (<= t_2 0.0) (fma (/ x z) (- y a) t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((y - z) / (a - z)), (t - x), x);
	double t_2 = x + ((y - z) * ((t - x) / (a - z)));
	double tmp;
	if (t_2 <= -5e-243) {
		tmp = t_1;
	} else if (t_2 <= 0.0) {
		tmp = fma((x / z), (y - a), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(y - z) / Float64(a - z)), Float64(t - x), x)
	t_2 = Float64(x + Float64(Float64(y - z) * Float64(Float64(t - x) / Float64(a - z))))
	tmp = 0.0
	if (t_2 <= -5e-243)
		tmp = t_1;
	elseif (t_2 <= 0.0)
		tmp = fma(Float64(x / z), Float64(y - a), t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision]}, Block[{t$95$2 = N[(x + N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e-243], t$95$1, If[LessEqual[t$95$2, 0.0], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -5e-243 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 91.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{\left(t - x\right)}, x\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), \left(\color{blue}{t} - x\right), x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), \left(t - x\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \left(t - x\right), x\right) \]

      14. --lowering--.f64 97.2%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 97.2%

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

   if -5e-243 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 5.3%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 96.8%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, \mathsf{\_.f64}\left(y, a\right), t\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 96.8%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{\_.f64}\left(\color{blue}{y}, a\right), t\right) \]

   8. Simplified 96.8%

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

Alternative 2: 69.7% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{z}{a - z}, x\right)\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;z \leq 1.72 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.55e+158)
   (fma (/ x z) (- y a) t)
   (if (<= z -4e+79)
     (fma (- x t) (/ z (- a z)) x)
     (if (<= z -5e-99)
       (* t (/ (- y z) (- a z)))
       (if (<= z 1.72e-22) (fma (/ y a) (- t x) x) (fma (/ (- x t) z) y t))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.55e+158) {
		tmp = fma((x / z), (y - a), t);
	} else if (z <= -4e+79) {
		tmp = fma((x - t), (z / (a - z)), x);
	} else if (z <= -5e-99) {
		tmp = t * ((y - z) / (a - z));
	} else if (z <= 1.72e-22) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.55e+158)
		tmp = fma(Float64(x / z), Float64(y - a), t);
	elseif (z <= -4e+79)
		tmp = fma(Float64(x - t), Float64(z / Float64(a - z)), x);
	elseif (z <= -5e-99)
		tmp = Float64(t * Float64(Float64(y - z) / Float64(a - z)));
	elseif (z <= 1.72e-22)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.55e+158], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, -4e+79], N[(N[(x - t), $MachinePrecision] * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, -5e-99], N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.72e-22], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]]]]

Derivation

1. Split input into 5 regimes

2. if z < -1.5500000000000001e158

   1. Initial program 76.2%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 81.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, \mathsf{\_.f64}\left(y, a\right), t\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 78.3%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{\_.f64}\left(\color{blue}{y}, a\right), t\right) \]

   8. Simplified 78.3%

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

   if -1.5500000000000001e158 < z < -3.99999999999999987e79

   1. Initial program 99.8%

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

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

      7. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \left(t - x\right)\right), \color{blue}{\left(\frac{z}{a - z}\right)}, x\right) \]

      8. mul-1-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(t - x\right)\right)\right), \left(\frac{\color{blue}{z}}{a - z}\right), x\right) \]

      9. sub-neg N/A

         \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)\right)\right), \left(\frac{z}{a - z}\right), x\right) \]

      10. +-commutative N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(x\right)\right) + t\right)\right)\right), \left(\frac{z}{a - z}\right), x\right) \]

      11. distribute-neg-in N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) + \left(\mathsf{neg}\left(t\right)\right)\right), \left(\frac{\color{blue}{z}}{a - z}\right), x\right) \]

      12. unsub-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(x\right)\right)\right)\right) - t\right), \left(\frac{\color{blue}{z}}{a - z}\right), x\right) \]

      13. remove-double-neg N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(x - t\right), \left(\frac{z}{a - z}\right), x\right) \]

      14. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, t\right), \left(\frac{\color{blue}{z}}{a - z}\right), x\right) \]

      15. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, t\right), \mathsf{/.f64}\left(z, \color{blue}{\left(a - z\right)}\right), x\right) \]

      16. --lowering--.f64 82.0%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{\_.f64}\left(x, t\right), \mathsf{/.f64}\left(z, \mathsf{\_.f64}\left(a, \color{blue}{z}\right)\right), x\right) \]

   5. Simplified 82.0%

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

   if -3.99999999999999987e79 < z < -4.99999999999999969e-99

   1. Initial program 73.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(t \cdot \left(y - z\right)\right), \color{blue}{\left(a - z\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(y - z\right)\right), \left(\color{blue}{a} - z\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(y, z\right)\right), \left(a - z\right)\right) \]

      4. --lowering--.f64 66.5%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(a, \color{blue}{z}\right)\right) \]

   5. Simplified 66.5%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), t\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), t\right) \]

      6. --lowering--.f64 68.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), t\right) \]

   7. Applied egg-rr 68.8%

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

   if -4.99999999999999969e-99 < z < 1.72000000000000001e-22

   1. Initial program 86.5%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{\left(t - x\right)}, x\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), \left(\color{blue}{t} - x\right), x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), \left(t - x\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \left(t - x\right), x\right) \]

      14. --lowering--.f64 95.9%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{y}{a}\right)}, \mathsf{\_.f64}\left(t, x\right), x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 83.4%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, a\right), \mathsf{\_.f64}\left(\color{blue}{t}, x\right), x\right) \]

   7. Simplified 83.4%

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

   if 1.72000000000000001e-22 < z

   1. Initial program 73.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 81.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, t\right), z\right), \color{blue}{y}, t\right) \]
   7. Step-by-step derivation

   8. Simplified 78.8%

      \[\leadsto \mathsf{fma}\left(\frac{x - t}{z}, \color{blue}{y}, t\right) \]
3. Recombined 5 regimes into one program.

4. Final simplification 79.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.55 \cdot 10^{+158}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq -4 \cdot 10^{+79}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{z}{a - z}, x\right)\\ \mathbf{elif}\;z \leq -5 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;z \leq 1.72 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 79.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -1.25 \cdot 10^{+14}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -6.5 \cdot 10^{-100}:\\ \;\;\;\;x + \left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 2 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) z) (- y a) t)))
   (if (<= z -1.25e+14)
     t_1
     (if (<= z -6.5e-100)
       (+ x (* (- y z) (/ t (- a z))))
       (if (<= z 2e-10) (fma (/ y (- a z)) (- t x) x) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / z), (y - a), t);
	double tmp;
	if (z <= -1.25e+14) {
		tmp = t_1;
	} else if (z <= -6.5e-100) {
		tmp = x + ((y - z) * (t / (a - z)));
	} else if (z <= 2e-10) {
		tmp = fma((y / (a - z)), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -1.25e+14)
		tmp = t_1;
	elseif (z <= -6.5e-100)
		tmp = Float64(x + Float64(Float64(y - z) * Float64(t / Float64(a - z))));
	elseif (z <= 2e-10)
		tmp = fma(Float64(y / Float64(a - z)), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -1.25e+14], t$95$1, If[LessEqual[z, -6.5e-100], N[(x + N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2e-10], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.25e14 or 2.00000000000000007e-10 < z

   1. Initial program 73.1%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 79.6%

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

   if -1.25e14 < z < -6.50000000000000013e-100

   1. Initial program 93.1%

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

   3. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, z\right), \color{blue}{\left(\frac{t}{a - z}\right)}\right)\right) \]
   4. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{/.f64}\left(t, \color{blue}{\left(a - z\right)}\right)\right)\right) \]

      2. --lowering--.f64 82.6%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{/.f64}\left(t, \mathsf{\_.f64}\left(a, \color{blue}{z}\right)\right)\right)\right) \]

   5. Simplified 82.6%

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

   if -6.50000000000000013e-100 < z < 2.00000000000000007e-10

   1. Initial program 86.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{\left(t - x\right)}, x\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), \left(\color{blue}{t} - x\right), x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), \left(t - x\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \left(t - x\right), x\right) \]

      14. --lowering--.f64 96.0%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 96.0%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{y}{a - z}\right)}, \mathsf{\_.f64}\left(t, x\right), x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, \left(a - z\right)\right), \mathsf{\_.f64}\left(\color{blue}{t}, x\right), x\right) \]

      2. --lowering--.f64 90.2%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, x\right), x\right) \]

   7. Simplified 90.2%

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

Alternative 4: 69.9% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -6.2e+72)
   (fma (/ x z) (- y a) t)
   (if (<= z -2.6e-99)
     (* t (/ (- y z) (- a z)))
     (if (<= z 1.55e-22) (fma (/ y a) (- t x) x) (fma (/ (- x t) z) y t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -6.2e+72) {
		tmp = fma((x / z), (y - a), t);
	} else if (z <= -2.6e-99) {
		tmp = t * ((y - z) / (a - z));
	} else if (z <= 1.55e-22) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -6.2e+72)
		tmp = fma(Float64(x / z), Float64(y - a), t);
	elseif (z <= -2.6e-99)
		tmp = Float64(t * Float64(Float64(y - z) / Float64(a - z)));
	elseif (z <= 1.55e-22)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -6.2e+72], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, -2.6e-99], N[(t * N[(N[(y - z), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.55e-22], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -6.19999999999999977e72

   1. Initial program 81.1%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 74.4%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, \mathsf{\_.f64}\left(y, a\right), t\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 69.8%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{\_.f64}\left(\color{blue}{y}, a\right), t\right) \]

   8. Simplified 69.8%

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

   if -6.19999999999999977e72 < z < -2.60000000000000005e-99

   1. Initial program 77.1%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(t \cdot \left(y - z\right)\right), \color{blue}{\left(a - z\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(y - z\right)\right), \left(\color{blue}{a} - z\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(y, z\right)\right), \left(a - z\right)\right) \]

      4. --lowering--.f64 67.1%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(a, \color{blue}{z}\right)\right) \]

   5. Simplified 67.1%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), t\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), t\right) \]

      6. --lowering--.f64 69.5%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), t\right) \]

   7. Applied egg-rr 69.5%

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

   if -2.60000000000000005e-99 < z < 1.55000000000000006e-22

   1. Initial program 86.5%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{\left(t - x\right)}, x\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), \left(\color{blue}{t} - x\right), x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), \left(t - x\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \left(t - x\right), x\right) \]

      14. --lowering--.f64 95.9%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{y}{a}\right)}, \mathsf{\_.f64}\left(t, x\right), x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 83.4%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, a\right), \mathsf{\_.f64}\left(\color{blue}{t}, x\right), x\right) \]

   7. Simplified 83.4%

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

   if 1.55000000000000006e-22 < z

   1. Initial program 73.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 81.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, t\right), z\right), \color{blue}{y}, t\right) \]
   7. Step-by-step derivation

   8. Simplified 78.8%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -6.2 \cdot 10^{+72}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq -2.6 \cdot 10^{-99}:\\ \;\;\;\;t \cdot \frac{y - z}{a - z}\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 69.6% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -5 \cdot 10^{+71}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq -4.2 \cdot 10^{-99}:\\ \;\;\;\;\left(y - z\right) \cdot \frac{t}{a - z}\\ \mathbf{elif}\;z \leq 2.55 \cdot 10^{-23}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -5e+71)
   (fma (/ x z) (- y a) t)
   (if (<= z -4.2e-99)
     (* (- y z) (/ t (- a z)))
     (if (<= z 2.55e-23) (fma (/ y a) (- t x) x) (fma (/ (- x t) z) y t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -5e+71) {
		tmp = fma((x / z), (y - a), t);
	} else if (z <= -4.2e-99) {
		tmp = (y - z) * (t / (a - z));
	} else if (z <= 2.55e-23) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -5e+71)
		tmp = fma(Float64(x / z), Float64(y - a), t);
	elseif (z <= -4.2e-99)
		tmp = Float64(Float64(y - z) * Float64(t / Float64(a - z)));
	elseif (z <= 2.55e-23)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -5e+71], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, -4.2e-99], N[(N[(y - z), $MachinePrecision] * N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 2.55e-23], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -4.99999999999999972e71

   1. Initial program 79.6%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 74.9%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, \mathsf{\_.f64}\left(y, a\right), t\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 70.4%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{\_.f64}\left(\color{blue}{y}, a\right), t\right) \]

   8. Simplified 70.4%

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

   if -4.99999999999999972e71 < z < -4.19999999999999968e-99

   1. Initial program 78.9%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(t \cdot \left(y - z\right)\right), \color{blue}{\left(a - z\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(y - z\right)\right), \left(\color{blue}{a} - z\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(y, z\right)\right), \left(a - z\right)\right) \]

      4. --lowering--.f64 66.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(a, \color{blue}{z}\right)\right) \]

   5. Simplified 66.3%

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(y - z\right), \color{blue}{\left(\frac{t}{a - z}\right)}\right) \]

      4. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(\frac{\color{blue}{t}}{a - z}\right)\right) \]

      5. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{/.f64}\left(t, \color{blue}{\left(a - z\right)}\right)\right) \]

      6. --lowering--.f64 68.8%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{/.f64}\left(t, \mathsf{\_.f64}\left(a, \color{blue}{z}\right)\right)\right) \]

   7. Applied egg-rr 68.8%

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

   if -4.19999999999999968e-99 < z < 2.55000000000000005e-23

   1. Initial program 86.5%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{\left(t - x\right)}, x\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), \left(\color{blue}{t} - x\right), x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), \left(t - x\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \left(t - x\right), x\right) \]

      14. --lowering--.f64 95.9%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 95.9%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{y}{a}\right)}, \mathsf{\_.f64}\left(t, x\right), x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 83.4%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, a\right), \mathsf{\_.f64}\left(\color{blue}{t}, x\right), x\right) \]

   7. Simplified 83.4%

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

   if 2.55000000000000005e-23 < z

   1. Initial program 73.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 81.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, t\right), z\right), \color{blue}{y}, t\right) \]
   7. Step-by-step derivation

   8. Simplified 78.8%

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

Alternative 6: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.25 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) z) (- y a) t)))
   (if (<= z -5.2e-44)
     t_1
     (if (<= z 1.25e-10) (fma (/ y (- a z)) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / z), (y - a), t);
	double tmp;
	if (z <= -5.2e-44) {
		tmp = t_1;
	} else if (z <= 1.25e-10) {
		tmp = fma((y / (a - z)), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -5.2e-44)
		tmp = t_1;
	elseif (z <= 1.25e-10)
		tmp = fma(Float64(y / Float64(a - z)), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -5.2e-44], t$95$1, If[LessEqual[z, 1.25e-10], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999996e-44 or 1.25000000000000008e-10 < z

   1. Initial program 75.4%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 78.2%

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

   if -5.1999999999999996e-44 < z < 1.25000000000000008e-10

   1. Initial program 86.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{\left(t - x\right)}, x\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), \left(\color{blue}{t} - x\right), x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), \left(t - x\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \left(t - x\right), x\right) \]

      14. --lowering--.f64 95.7%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 95.7%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{y}{a - z}\right)}, \mathsf{\_.f64}\left(t, x\right), x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, \left(a - z\right)\right), \mathsf{\_.f64}\left(\color{blue}{t}, x\right), x\right) \]

      2. --lowering--.f64 87.7%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, x\right), x\right) \]

   7. Simplified 87.7%

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

Alternative 7: 76.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.35 \cdot 10^{+14}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-10}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a - z}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1.35e+14)
   (fma (/ x z) (- y a) t)
   (if (<= z 2.6e-10) (fma (/ y (- a z)) (- t x) x) (fma (/ (- x t) z) y t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1.35e+14) {
		tmp = fma((x / z), (y - a), t);
	} else if (z <= 2.6e-10) {
		tmp = fma((y / (a - z)), (t - x), x);
	} else {
		tmp = fma(((x - t) / z), y, t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1.35e+14)
		tmp = fma(Float64(x / z), Float64(y - a), t);
	elseif (z <= 2.6e-10)
		tmp = fma(Float64(y / Float64(a - z)), Float64(t - x), x);
	else
		tmp = fma(Float64(Float64(x - t) / z), y, t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1.35e+14], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 2.6e-10], N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -1.35e14

   1. Initial program 73.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 76.5%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, \mathsf{\_.f64}\left(y, a\right), t\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 69.8%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{\_.f64}\left(\color{blue}{y}, a\right), t\right) \]

   8. Simplified 69.8%

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

   if -1.35e14 < z < 2.59999999999999981e-10

   1. Initial program 87.6%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{\left(t - x\right)}, x\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), \left(\color{blue}{t} - x\right), x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), \left(t - x\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \left(t - x\right), x\right) \]

      14. --lowering--.f64 95.4%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 95.4%

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

   5. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{y}{a - z}\right)}, \mathsf{\_.f64}\left(t, x\right), x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, \left(a - z\right)\right), \mathsf{\_.f64}\left(\color{blue}{t}, x\right), x\right) \]

      2. --lowering--.f64 84.3%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, x\right), x\right) \]

   7. Simplified 84.3%

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

   if 2.59999999999999981e-10 < z

   1. Initial program 73.1%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 83.3%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, t\right), z\right), \color{blue}{y}, t\right) \]
   7. Step-by-step derivation

   8. Simplified 80.5%

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

Alternative 8: 69.5% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x - t}{z}, y, t\right)\\ \mathbf{if}\;z \leq -4 \cdot 10^{-75}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.01 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- x t) z) y t)))
   (if (<= z -4e-75) t_1 (if (<= z 1.01e-22) (fma (/ y a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((x - t) / z), y, t);
	double tmp;
	if (z <= -4e-75) {
		tmp = t_1;
	} else if (z <= 1.01e-22) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(x - t) / z), y, t)
	tmp = 0.0
	if (z <= -4e-75)
		tmp = t_1;
	elseif (z <= 1.01e-22)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * y + t), $MachinePrecision]}, If[LessEqual[z, -4e-75], t$95$1, If[LessEqual[z, 1.01e-22], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.9999999999999998e-75 or 1.01e-22 < z

   1. Initial program 77.0%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 76.1%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, t\right), z\right), \color{blue}{y}, t\right) \]
   7. Step-by-step derivation

   8. Simplified 70.8%

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

   if -3.9999999999999998e-75 < z < 1.01e-22

   1. Initial program 86.2%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{\left(t - x\right)}, x\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), \left(\color{blue}{t} - x\right), x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), \left(t - x\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \left(t - x\right), x\right) \]

      14. --lowering--.f64 95.2%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 95.2%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{y}{a}\right)}, \mathsf{\_.f64}\left(t, x\right), x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 82.4%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, a\right), \mathsf{\_.f64}\left(\color{blue}{t}, x\right), x\right) \]

   7. Simplified 82.4%

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

Alternative 9: 69.7% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -5 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.6 \cdot 10^{-7}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x z) (- y a) t)))
   (if (<= z -5e-44) t_1 (if (<= z 2.6e-7) (fma (/ y a) (- t x) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x / z), (y - a), t);
	double tmp;
	if (z <= -5e-44) {
		tmp = t_1;
	} else if (z <= 2.6e-7) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -5e-44)
		tmp = t_1;
	elseif (z <= 2.6e-7)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -5e-44], t$95$1, If[LessEqual[z, 2.6e-7], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.00000000000000039e-44 or 2.59999999999999999e-7 < z

   1. Initial program 75.2%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 78.1%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, \mathsf{\_.f64}\left(y, a\right), t\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 69.7%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{\_.f64}\left(\color{blue}{y}, a\right), t\right) \]

   8. Simplified 69.7%

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

   if -5.00000000000000039e-44 < z < 2.59999999999999999e-7

   1. Initial program 87.0%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{t - x}} + x \]

      6. flip3-- N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{1}{\frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)}}} + x \]

      7. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{1}{\frac{t \cdot t + \left(x \cdot x + t \cdot x\right)}{{t}^{3} - {x}^{3}}} + x \]

      8. clear-num N/A

         \[\leadsto \frac{y - z}{a - z} \cdot \frac{{t}^{3} - {x}^{3}}{t \cdot t + \left(x \cdot x + t \cdot x\right)} + x \]

      9. flip3-- N/A

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

      10. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{\left(t - x\right)}, x\right) \]

      11. /-lowering-/.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), \left(\color{blue}{t} - x\right), x\right) \]

      12. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), \left(t - x\right), x\right) \]

      13. --lowering--.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \left(t - x\right), x\right) \]

      14. --lowering--.f64 95.7%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), \mathsf{\_.f64}\left(t, \color{blue}{x}\right), x\right) \]

   4. Applied egg-rr 95.7%

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

   5. Taylor expanded in z around 0

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{y}{a}\right)}, \mathsf{\_.f64}\left(t, x\right), x\right) \]
   6. Step-by-step derivation

      1. /-lowering-/.f64 78.0%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(y, a\right), \mathsf{\_.f64}\left(\color{blue}{t}, x\right), x\right) \]

   7. Simplified 78.0%

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

Alternative 10: 68.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -5.2 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.15 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x z) (- y a) t)))
   (if (<= z -5.2e-44) t_1 (if (<= z 1.15e-22) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x / z), (y - a), t);
	double tmp;
	if (z <= -5.2e-44) {
		tmp = t_1;
	} else if (z <= 1.15e-22) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x / z), Float64(y - a), t)
	tmp = 0.0
	if (z <= -5.2e-44)
		tmp = t_1;
	elseif (z <= 1.15e-22)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]}, If[LessEqual[z, -5.2e-44], t$95$1, If[LessEqual[z, 1.15e-22], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -5.1999999999999996e-44 or 1.1499999999999999e-22 < z

   1. Initial program 75.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 77.6%

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

   6. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, \mathsf{\_.f64}\left(y, a\right), t\right) \]
   7. Step-by-step derivation

      1. /-lowering-/.f64 68.2%

         \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, z\right), \mathsf{\_.f64}\left(\color{blue}{y}, a\right), t\right) \]

   8. Simplified 68.2%

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

   if -5.1999999999999996e-44 < z < 1.1499999999999999e-22

   1. Initial program 87.1%

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

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{\left(\frac{t - x}{a}\right)}, x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{/.f64}\left(\left(t - x\right), \color{blue}{a}\right), x\right) \]

      5. --lowering--.f64 72.6%

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, x\right), a\right), x\right) \]

   5. Simplified 72.6%

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

Alternative 11: 64.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{if}\;z \leq -3.5 \cdot 10^{-44}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 2.35 \cdot 10^{-22}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x z) y t)))
   (if (<= z -3.5e-44) t_1 (if (<= z 2.35e-22) (fma y (/ (- t x) a) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((x / z), y, t);
	double tmp;
	if (z <= -3.5e-44) {
		tmp = t_1;
	} else if (z <= 2.35e-22) {
		tmp = fma(y, ((t - x) / a), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(x / z), y, t)
	tmp = 0.0
	if (z <= -3.5e-44)
		tmp = t_1;
	elseif (z <= 2.35e-22)
		tmp = fma(y, Float64(Float64(t - x) / a), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(x / z), $MachinePrecision] * y + t), $MachinePrecision]}, If[LessEqual[z, -3.5e-44], t$95$1, If[LessEqual[z, 2.35e-22], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.4999999999999998e-44 or 2.3500000000000001e-22 < z

   1. Initial program 75.7%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 77.6%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, t\right), z\right), \color{blue}{y}, t\right) \]
   7. Step-by-step derivation

   8. Simplified 71.9%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, y, t\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 62.8%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, z\right), y, t\right) \]

   11. Simplified 62.8%

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

   if -3.4999999999999998e-44 < z < 2.3500000000000001e-22

   1. Initial program 87.1%

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

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

      2. associate-/l* N/A

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

      3. accelerator-lowering-fma.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \color{blue}{\left(\frac{t - x}{a}\right)}, x\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{/.f64}\left(\left(t - x\right), \color{blue}{a}\right), x\right) \]

      5. --lowering--.f64 72.6%

         \[\leadsto \mathsf{fma.f64}\left(y, \mathsf{/.f64}\left(\mathsf{\_.f64}\left(t, x\right), a\right), x\right) \]

   5. Simplified 72.6%

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

Alternative 12: 47.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-6}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-70}:\\ \;\;\;\;t \cdot \frac{z - y}{z}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+206}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1e-6)
   (+ x t)
   (if (<= a 2.2e-70) (* t (/ (- z y) z)) (if (<= a 1.9e+206) (+ x t) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1e-6) {
		tmp = x + t;
	} else if (a <= 2.2e-70) {
		tmp = t * ((z - y) / z);
	} else if (a <= 1.9e+206) {
		tmp = x + t;
	} else {
		tmp = x;
	}
	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 (a <= (-1d-6)) then
        tmp = x + t
    else if (a <= 2.2d-70) then
        tmp = t * ((z - y) / z)
    else if (a <= 1.9d+206) then
        tmp = x + t
    else
        tmp = x
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1e-6) {
		tmp = x + t;
	} else if (a <= 2.2e-70) {
		tmp = t * ((z - y) / z);
	} else if (a <= 1.9e+206) {
		tmp = x + t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -1e-6:
		tmp = x + t
	elif a <= 2.2e-70:
		tmp = t * ((z - y) / z)
	elif a <= 1.9e+206:
		tmp = x + t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1e-6)
		tmp = Float64(x + t);
	elseif (a <= 2.2e-70)
		tmp = Float64(t * Float64(Float64(z - y) / z));
	elseif (a <= 1.9e+206)
		tmp = Float64(x + t);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -1e-6)
		tmp = x + t;
	elseif (a <= 2.2e-70)
		tmp = t * ((z - y) / z);
	elseif (a <= 1.9e+206)
		tmp = x + t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1e-6], N[(x + t), $MachinePrecision], If[LessEqual[a, 2.2e-70], N[(t * N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[a, 1.9e+206], N[(x + t), $MachinePrecision], x]]]

Derivation

1. Split input into 3 regimes

2. if a < -9.99999999999999955e-7 or 2.1999999999999999e-70 < a < 1.8999999999999999e206

   1. Initial program 90.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(t - x\right)}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 18.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{x}\right)\right) \]

   5. Simplified 18.4%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{t}\right) \]
   7. Step-by-step derivation

   8. Simplified 47.7%

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

   if -9.99999999999999955e-7 < a < 2.1999999999999999e-70

   1. Initial program 71.7%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(t \cdot \left(y - z\right)\right), \color{blue}{\left(a - z\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(y - z\right)\right), \left(\color{blue}{a} - z\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(y, z\right)\right), \left(a - z\right)\right) \]

      4. --lowering--.f64 55.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(a, \color{blue}{z}\right)\right) \]

   5. Simplified 55.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), t\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), t\right) \]

      6. --lowering--.f64 66.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), t\right) \]

   7. Applied egg-rr 66.9%

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

   8. Taylor expanded in a around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(-1 \cdot \frac{y - z}{z}\right)}, t\right) \]
   9. Step-by-step derivation

      1. associate-*r/ N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{-1 \cdot \left(y - z\right)}{z}\right), t\right) \]

      2. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(-1 \cdot \left(y - z\right)\right), z\right), t\right) \]

      3. mul-1-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y - z\right)\right)\right), z\right), t\right) \]

      4. sub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(y + \left(\mathsf{neg}\left(z\right)\right)\right)\right)\right), z\right), t\right) \]

      5. +-commutative N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\mathsf{neg}\left(\left(\left(\mathsf{neg}\left(z\right)\right) + y\right)\right)\right), z\right), t\right) \]

      6. distribute-neg-in N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) + \left(\mathsf{neg}\left(y\right)\right)\right), z\right), t\right) \]

      7. unsub-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(\left(\mathsf{neg}\left(\left(\mathsf{neg}\left(z\right)\right)\right)\right) - y\right), z\right), t\right) \]

      8. remove-double-neg N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(z - y\right), z\right), t\right) \]

      9. --lowering--.f64 61.9%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(z, y\right), z\right), t\right) \]

   10. Simplified 61.9%

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

   if 1.8999999999999999e206 < a

   1. Initial program 88.2%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 70.6%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.

4. Final simplification 56.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1 \cdot 10^{-6}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;a \leq 2.2 \cdot 10^{-70}:\\ \;\;\;\;t \cdot \frac{z - y}{z}\\ \mathbf{elif}\;a \leq 1.9 \cdot 10^{+206}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 50.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -1.25 \cdot 10^{-11}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;a \leq 1.2 \cdot 10^{+31}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -1.25e-11) (+ x t) (if (<= a 1.2e+31) (fma (/ x z) y t) x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -1.25e-11) {
		tmp = x + t;
	} else if (a <= 1.2e+31) {
		tmp = fma((x / z), y, t);
	} else {
		tmp = x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -1.25e-11)
		tmp = Float64(x + t);
	elseif (a <= 1.2e+31)
		tmp = fma(Float64(x / z), y, t);
	else
		tmp = x;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -1.25e-11], N[(x + t), $MachinePrecision], If[LessEqual[a, 1.2e+31], N[(N[(x / z), $MachinePrecision] * y + t), $MachinePrecision], x]]

Derivation

1. Split input into 3 regimes

2. if a < -1.25000000000000005e-11

   1. Initial program 98.0%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(t - x\right)}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 17.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{x}\right)\right) \]

   5. Simplified 17.0%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{t}\right) \]
   7. Step-by-step derivation

   8. Simplified 52.4%

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

   if -1.25000000000000005e-11 < a < 1.19999999999999991e31

   1. Initial program 72.8%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

      12. accelerator-lowering-fma.f64 N/A

          \[\leadsto \mathsf{fma.f64}\left(\left(-1 \cdot \frac{t - x}{z}\right), \color{blue}{\left(y - a\right)}, t\right) \]

   5. Simplified 76.4%

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

   6. Taylor expanded in y around inf

      \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(x, t\right), z\right), \color{blue}{y}, t\right) \]
   7. Step-by-step derivation

   8. Simplified 71.7%

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

   9. Taylor expanded in x around inf

      \[\leadsto \mathsf{fma.f64}\left(\color{blue}{\left(\frac{x}{z}\right)}, y, t\right) \]
   10. Step-by-step derivation

       1. /-lowering-/.f64 55.9%

          \[\leadsto \mathsf{fma.f64}\left(\mathsf{/.f64}\left(x, z\right), y, t\right) \]

   11. Simplified 55.9%

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

   if 1.19999999999999991e31 < a

   1. Initial program 85.1%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 51.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 14: 39.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := t \cdot \frac{y}{a}\\ \mathbf{if}\;y \leq -2.05 \cdot 10^{+149}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+85}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (* t (/ y a))))
   (if (<= y -2.05e+149) t_1 (if (<= y 7.6e+85) (+ x t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (y / a);
	double tmp;
	if (y <= -2.05e+149) {
		tmp = t_1;
	} else if (y <= 7.6e+85) {
		tmp = x + t;
	} else {
		tmp = t_1;
	}
	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) :: t_1
    real(8) :: tmp
    t_1 = t * (y / a)
    if (y <= (-2.05d+149)) then
        tmp = t_1
    else if (y <= 7.6d+85) then
        tmp = x + t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double t_1 = t * (y / a);
	double tmp;
	if (y <= -2.05e+149) {
		tmp = t_1;
	} else if (y <= 7.6e+85) {
		tmp = x + t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	t_1 = t * (y / a)
	tmp = 0
	if y <= -2.05e+149:
		tmp = t_1
	elif y <= 7.6e+85:
		tmp = x + t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t, a)
	t_1 = Float64(t * Float64(y / a))
	tmp = 0.0
	if (y <= -2.05e+149)
		tmp = t_1;
	elseif (y <= 7.6e+85)
		tmp = Float64(x + t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	t_1 = t * (y / a);
	tmp = 0.0;
	if (y <= -2.05e+149)
		tmp = t_1;
	elseif (y <= 7.6e+85)
		tmp = x + t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(t * N[(y / a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[y, -2.05e+149], t$95$1, If[LessEqual[y, 7.6e+85], N[(x + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if y < -2.0499999999999998e149 or 7.59999999999999984e85 < y

   1. Initial program 89.0%

      \[x + \left(y - z\right) \cdot \frac{t - x}{a - z} \]
   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. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\left(t \cdot \left(y - z\right)\right), \color{blue}{\left(a - z\right)}\right) \]

      2. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \left(y - z\right)\right), \left(\color{blue}{a} - z\right)\right) \]

      3. --lowering--.f64 N/A

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(y, z\right)\right), \left(a - z\right)\right) \]

      4. --lowering--.f64 46.3%

         \[\leadsto \mathsf{/.f64}\left(\mathsf{*.f64}\left(t, \mathsf{\_.f64}\left(y, z\right)\right), \mathsf{\_.f64}\left(a, \color{blue}{z}\right)\right) \]

   5. Simplified 46.3%

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

      1. associate-/l* N/A

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

      2. *-commutative N/A

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

      3. *-lowering-*.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\left(\frac{y - z}{a - z}\right), \color{blue}{t}\right) \]

      4. /-lowering-/.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\left(y - z\right), \left(a - z\right)\right), t\right) \]

      5. --lowering--.f64 N/A

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \left(a - z\right)\right), t\right) \]

      6. --lowering--.f64 55.1%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(\mathsf{\_.f64}\left(y, z\right), \mathsf{\_.f64}\left(a, z\right)\right), t\right) \]

   7. Applied egg-rr 55.1%

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

   8. Taylor expanded in z around 0

      \[\leadsto \mathsf{*.f64}\left(\color{blue}{\left(\frac{y}{a}\right)}, t\right) \]
   9. Step-by-step derivation

      1. /-lowering-/.f64 38.0%

         \[\leadsto \mathsf{*.f64}\left(\mathsf{/.f64}\left(y, a\right), t\right) \]

   10. Simplified 38.0%

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

   if -2.0499999999999998e149 < y < 7.59999999999999984e85

   1. Initial program 77.9%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(t - x\right)}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 29.4%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{x}\right)\right) \]

   5. Simplified 29.4%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{t}\right) \]
   7. Step-by-step derivation

   8. Simplified 53.0%

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

4. Final simplification 48.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;y \leq -2.05 \cdot 10^{+149}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \mathbf{elif}\;y \leq 7.6 \cdot 10^{+85}:\\ \;\;\;\;x + t\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{y}{a}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 37.4% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{-75}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 17:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.5e-75) (+ x t) (if (<= z 17.0) x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.5e-75) {
		tmp = x + t;
	} else if (z <= 17.0) {
		tmp = x;
	} else {
		tmp = t;
	}
	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 <= (-3.5d-75)) then
        tmp = x + t
    else if (z <= 17.0d0) then
        tmp = x
    else
        tmp = t
    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 <= -3.5e-75) {
		tmp = x + t;
	} else if (z <= 17.0) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.5e-75:
		tmp = x + t
	elif z <= 17.0:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.5e-75)
		tmp = Float64(x + t);
	elseif (z <= 17.0)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.5e-75)
		tmp = x + t;
	elseif (z <= 17.0)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.5e-75], N[(x + t), $MachinePrecision], If[LessEqual[z, 17.0], x, t]]

Derivation

1. Split input into 3 regimes

2. if z < -3.49999999999999985e-75

   1. Initial program 79.2%

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

   3. Taylor expanded in z around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{\left(t - x\right)}\right) \]
   4. Step-by-step derivation

      1. --lowering--.f64 33.0%

         \[\leadsto \mathsf{+.f64}\left(x, \mathsf{\_.f64}\left(t, \color{blue}{x}\right)\right) \]

   5. Simplified 33.0%

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

   6. Taylor expanded in t around inf

      \[\leadsto \mathsf{+.f64}\left(x, \color{blue}{t}\right) \]
   7. Step-by-step derivation

   8. Simplified 46.0%

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

   if -3.49999999999999985e-75 < z < 17

   1. Initial program 84.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 39.7%

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

   if 17 < z

   1. Initial program 74.9%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 55.7%

      \[\leadsto \color{blue}{t} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 16: 37.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{-76}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 15.5:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e-76) t (if (<= z 15.5) x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e-76) {
		tmp = t;
	} else if (z <= 15.5) {
		tmp = x;
	} else {
		tmp = t;
	}
	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 <= (-1d-76)) then
        tmp = t
    else if (z <= 15.5d0) then
        tmp = x
    else
        tmp = t
    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 <= -1e-76) {
		tmp = t;
	} else if (z <= 15.5) {
		tmp = x;
	} else {
		tmp = t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e-76:
		tmp = t
	elif z <= 15.5:
		tmp = x
	else:
		tmp = t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e-76)
		tmp = t;
	elseif (z <= 15.5)
		tmp = x;
	else
		tmp = t;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1e-76)
		tmp = t;
	elseif (z <= 15.5)
		tmp = x;
	else
		tmp = t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e-76], t, If[LessEqual[z, 15.5], x, t]]

Derivation

1. Split input into 2 regimes

2. if z < -9.99999999999999927e-77 or 15.5 < z

   1. Initial program 77.6%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 47.2%

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

   if -9.99999999999999927e-77 < z < 15.5

   1. Initial program 84.9%

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

   3. Taylor expanded in a around inf

      \[\leadsto \color{blue}{x} \]
   4. Step-by-step derivation

   5. Simplified 39.7%

      \[\leadsto \color{blue}{x} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 17: 25.5% accurate, 29.0× speedup

Math

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

FPCore

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

C

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

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 = t
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 81.1%

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

3. Taylor expanded in z around inf

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

5. Simplified 27.7%

   \[\leadsto \color{blue}{t} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024193 
(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)))))