Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 80.1% → 94.3%

Time: 14.6s

Alternatives: 15

Speedup: 0.3×

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 80.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 94.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{y - z}{a - z}, t - x, x\right)\\ t_2 := x + \left(y - z\right) \cdot \frac{t - x}{a - z}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{-239}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

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 -2e-239) 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 <= -2e-239) {
		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 <= -2e-239)
		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, -2e-239], 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)))) < -2.0000000000000002e-239 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 89.5%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 93.1

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

   4. Applied egg-rr 93.1%

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

   if -2.0000000000000002e-239 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.6%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

   5. Simplified 99.7%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 99.7

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

   8. Simplified 99.7%

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

Alternative 2: 68.4% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+228}:\\ \;\;\;\;t + \frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{z}{a - z}, x\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -9.5e+228)
   (+ t (* (/ y z) (- x t)))
   (if (<= z -3.7e-14)
     (fma (- x t) (/ z (- a z)) x)
     (if (<= z 3e+44) (fma (/ y a) (- t x) x) (fma (/ x z) (- y a) t)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -9.5e+228) {
		tmp = t + ((y / z) * (x - t));
	} else if (z <= -3.7e-14) {
		tmp = fma((x - t), (z / (a - z)), x);
	} else if (z <= 3e+44) {
		tmp = fma((y / a), (t - x), x);
	} else {
		tmp = fma((x / z), (y - a), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -9.5e+228)
		tmp = Float64(t + Float64(Float64(y / z) * Float64(x - t)));
	elseif (z <= -3.7e-14)
		tmp = fma(Float64(x - t), Float64(z / Float64(a - z)), x);
	elseif (z <= 3e+44)
		tmp = fma(Float64(y / a), Float64(t - x), x);
	else
		tmp = fma(Float64(x / z), Float64(y - a), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -9.5e+228], N[(t + N[(N[(y / z), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, -3.7e-14], N[(N[(x - t), $MachinePrecision] * N[(z / N[(a - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 3e+44], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], N[(N[(x / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision]]]]

Derivation

1. Split input into 4 regimes

2. if z < -9.50000000000000046e228

   1. Initial program 53.8%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 63.0

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

   4. Applied egg-rr 63.0%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 N/A

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

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

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

      12. --lowering--.f64 99.9

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

   7. Simplified 99.9%

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

   8. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 95.7

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

   10. Simplified 95.7%

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

   if -9.50000000000000046e228 < z < -3.70000000000000001e-14

   1. Initial program 86.1%

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

   3. Taylor expanded in y around 0

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

      1. +-commutative N/A

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

      2. mul-1-neg N/A

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

      3. *-commutative N/A

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

      4. associate-/l* N/A

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

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

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

      6. mul-1-neg N/A

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

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

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

      8. mul-1-neg N/A

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

      9. sub-neg N/A

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

      10. +-commutative N/A

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

      11. distribute-neg-in N/A

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

      12. unsub-neg N/A

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

      13. remove-double-neg N/A

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

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

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

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

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

      16. --lowering--.f64 69.3

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

   5. Simplified 69.3%

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

   if -3.70000000000000001e-14 < z < 2.99999999999999987e44

   1. Initial program 93.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 95.0

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

   4. Applied egg-rr 95.0%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 75.2

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

   7. Simplified 75.2%

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

   if 2.99999999999999987e44 < z

   1. Initial program 55.9%

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

   3. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

   5. Simplified 78.3%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 71.9

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

   8. Simplified 71.9%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -9.5 \cdot 10^{+228}:\\ \;\;\;\;t + \frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{elif}\;z \leq -3.7 \cdot 10^{-14}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{z}{a - z}, x\right)\\ \mathbf{elif}\;z \leq 3 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 69.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -2.5 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -2.7 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{elif}\;z \leq 6.2 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t - x, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x z) (- y a) t)))
   (if (<= z -2.5e+114)
     t_1
     (if (<= z -2.7e-129)
       (fma (- y z) (/ t a) x)
       (if (<= z 6.2e+44) (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 <= -2.5e+114) {
		tmp = t_1;
	} else if (z <= -2.7e-129) {
		tmp = fma((y - z), (t / a), x);
	} else if (z <= 6.2e+44) {
		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 <= -2.5e+114)
		tmp = t_1;
	elseif (z <= -2.7e-129)
		tmp = fma(Float64(y - z), Float64(t / a), x);
	elseif (z <= 6.2e+44)
		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, -2.5e+114], t$95$1, If[LessEqual[z, -2.7e-129], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 6.2e+44], N[(N[(y / a), $MachinePrecision] * N[(t - x), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.5e114 or 6.19999999999999991e44 < z

   1. Initial program 60.2%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

   5. Simplified 81.1%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 74.3

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

   8. Simplified 74.3%

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

   if -2.5e114 < z < -2.69999999999999999e-129

   1. Initial program 96.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 62.4

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

   5. Simplified 62.4%

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

   6. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 62.4

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

   8. Simplified 62.4%

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

   if -2.69999999999999999e-129 < z < 6.19999999999999991e44

   1. Initial program 92.3%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 94.5

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

   4. Applied egg-rr 94.5%

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

   5. Taylor expanded in z around 0

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

      1. /-lowering-/.f64 79.2

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

   7. Simplified 79.2%

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

Alternative 4: 68.5% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, y - a, t\right)\\ \mathbf{if}\;z \leq -2.4 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{elif}\;z \leq 1.95 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x z) (- y a) t)))
   (if (<= z -2.4e+114)
     t_1
     (if (<= z -3.2e-129)
       (fma (- y z) (/ t a) x)
       (if (<= z 1.95e+44) (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 <= -2.4e+114) {
		tmp = t_1;
	} else if (z <= -3.2e-129) {
		tmp = fma((y - z), (t / a), x);
	} else if (z <= 1.95e+44) {
		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 <= -2.4e+114)
		tmp = t_1;
	elseif (z <= -3.2e-129)
		tmp = fma(Float64(y - z), Float64(t / a), x);
	elseif (z <= 1.95e+44)
		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, -2.4e+114], t$95$1, If[LessEqual[z, -3.2e-129], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 1.95e+44], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.4e114 or 1.9500000000000001e44 < z

   1. Initial program 60.2%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

   5. Simplified 81.1%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 74.3

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

   8. Simplified 74.3%

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

   if -2.4e114 < z < -3.2000000000000003e-129

   1. Initial program 96.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 62.4

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

   5. Simplified 62.4%

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

   6. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 62.4

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

   8. Simplified 62.4%

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

   if -3.2000000000000003e-129 < z < 1.9500000000000001e44

   1. Initial program 92.3%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 78.0

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

   5. Simplified 78.0%

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

Alternative 5: 65.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{if}\;z \leq -2.05 \cdot 10^{+117}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq -3.2 \cdot 10^{-129}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{elif}\;z \leq 7.5 \cdot 10^{+85}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x z) y t)))
   (if (<= z -2.05e+117)
     t_1
     (if (<= z -3.2e-129)
       (fma (- y z) (/ t a) x)
       (if (<= z 7.5e+85) (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 <= -2.05e+117) {
		tmp = t_1;
	} else if (z <= -3.2e-129) {
		tmp = fma((y - z), (t / a), x);
	} else if (z <= 7.5e+85) {
		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 <= -2.05e+117)
		tmp = t_1;
	elseif (z <= -3.2e-129)
		tmp = fma(Float64(y - z), Float64(t / a), x);
	elseif (z <= 7.5e+85)
		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, -2.05e+117], t$95$1, If[LessEqual[z, -3.2e-129], N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[z, 7.5e+85], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.05e117 or 7.49999999999999942e85 < z

   1. Initial program 59.5%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

   5. Simplified 83.2%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 75.8

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

   8. Simplified 75.8%

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

   9. Taylor expanded in y around inf

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

   11. Simplified 68.0%

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

   if -2.05e117 < z < -3.2000000000000003e-129

   1. Initial program 96.1%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 62.4

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

   5. Simplified 62.4%

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

   6. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 62.4

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

   8. Simplified 62.4%

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

   if -3.2000000000000003e-129 < z < 7.49999999999999942e85

   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 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 75.2

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

   5. Simplified 75.2%

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

Alternative 6: 74.1% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.1 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y - a, t\right)\\ \mathbf{elif}\;z \leq 5.4 \cdot 10^{+44}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y - a}{z}, x - t, t\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.1e+114)
   (fma (/ (- x t) z) (- y a) t)
   (if (<= z 5.4e+44)
     (fma (- y z) (/ (- t x) a) x)
     (fma (/ (- y a) z) (- x t) t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.1e+114) {
		tmp = fma(((x - t) / z), (y - a), t);
	} else if (z <= 5.4e+44) {
		tmp = fma((y - z), ((t - x) / a), x);
	} else {
		tmp = fma(((y - a) / z), (x - t), t);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.1e+114)
		tmp = fma(Float64(Float64(x - t) / z), Float64(y - a), t);
	elseif (z <= 5.4e+44)
		tmp = fma(Float64(y - z), Float64(Float64(t - x) / a), x);
	else
		tmp = fma(Float64(Float64(y - a) / z), Float64(x - t), t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.1e+114], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[z, 5.4e+44], N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(y - a), $MachinePrecision] / z), $MachinePrecision] * N[(x - t), $MachinePrecision] + t), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.1e114

   1. Initial program 66.5%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

   5. Simplified 85.1%

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

   if -3.1e114 < z < 5.4e44

   1. Initial program 93.7%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 72.7

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

   5. Simplified 72.7%

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

   if 5.4e44 < z

   1. Initial program 55.9%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 59.9

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

   4. Applied egg-rr 59.9%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 N/A

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

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

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

      12. --lowering--.f64 82.6

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

   7. Simplified 82.6%

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

      1. cancel-sign-sub-inv N/A

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

      2. sub-neg N/A

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

      3. distribute-neg-in N/A

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

      4. remove-double-neg N/A

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

      5. +-commutative N/A

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

      6. sub-neg N/A

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

      7. associate-/l* N/A

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

      8. associate-*l/ N/A

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

      9. +-commutative N/A

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

      10. *-commutative N/A

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

   9. Applied egg-rr 82.6%

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

Alternative 7: 72.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -2.3 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.2 \cdot 10^{+68}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x - t}{z}, y - a, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -2.3e+114)
     t_1
     (if (<= a 6.2e+68) (fma (/ (- x t) 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), ((t - x) / a), x);
	double tmp;
	if (a <= -2.3e+114) {
		tmp = t_1;
	} else if (a <= 6.2e+68) {
		tmp = fma(((x - t) / z), (y - a), t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(Float64(t - x) / a), x)
	tmp = 0.0
	if (a <= -2.3e+114)
		tmp = t_1;
	elseif (a <= 6.2e+68)
		tmp = fma(Float64(Float64(x - t) / 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[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -2.3e+114], t$95$1, If[LessEqual[a, 6.2e+68], N[(N[(N[(x - t), $MachinePrecision] / z), $MachinePrecision] * N[(y - a), $MachinePrecision] + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.3e114 or 6.1999999999999997e68 < a

   1. Initial program 89.4%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 79.5

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

   5. Simplified 79.5%

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

   if -2.3e114 < a < 6.1999999999999997e68

   1. Initial program 72.4%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

   5. Simplified 74.6%

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

Alternative 8: 71.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{if}\;a \leq -1.22 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{+70}:\\ \;\;\;\;t + \frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ (- t x) a) x)))
   (if (<= a -1.22e+114)
     t_1
     (if (<= a 1.55e+70) (+ t (* (/ y z) (- x t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), ((t - x) / a), x);
	double tmp;
	if (a <= -1.22e+114) {
		tmp = t_1;
	} else if (a <= 1.55e+70) {
		tmp = t + ((y / z) * (x - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(Float64(t - x) / a), x)
	tmp = 0.0
	if (a <= -1.22e+114)
		tmp = t_1;
	elseif (a <= 1.55e+70)
		tmp = Float64(t + Float64(Float64(y / z) * Float64(x - t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.22e+114], t$95$1, If[LessEqual[a, 1.55e+70], N[(t + N[(N[(y / z), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.21999999999999999e114 or 1.55000000000000015e70 < a

   1. Initial program 89.4%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 79.5

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

   5. Simplified 79.5%

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

   if -1.21999999999999999e114 < a < 1.55000000000000015e70

   1. Initial program 72.4%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 75.5

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

   4. Applied egg-rr 75.5%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 N/A

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

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

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

      12. --lowering--.f64 75.7

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

   7. Simplified 75.7%

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

   8. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 70.9

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

   10. Simplified 70.9%

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

4. Final simplification 74.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.22 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \mathbf{elif}\;a \leq 1.55 \cdot 10^{+70}:\\ \;\;\;\;t + \frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t - x}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 68.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{if}\;a \leq -1.22 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;t + \frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (- y z) (/ t a) x)))
   (if (<= a -1.22e+114)
     t_1
     (if (<= a 7.2e+111) (+ t (* (/ y z) (- x t))) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((y - z), (t / a), x);
	double tmp;
	if (a <= -1.22e+114) {
		tmp = t_1;
	} else if (a <= 7.2e+111) {
		tmp = t + ((y / z) * (x - t));
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(y - z), Float64(t / a), x)
	tmp = 0.0
	if (a <= -1.22e+114)
		tmp = t_1;
	elseif (a <= 7.2e+111)
		tmp = Float64(t + Float64(Float64(y / z) * Float64(x - t)));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(y - z), $MachinePrecision] * N[(t / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.22e+114], t$95$1, If[LessEqual[a, 7.2e+111], N[(t + N[(N[(y / z), $MachinePrecision] * N[(x - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.21999999999999999e114 or 7.2000000000000004e111 < a

   1. Initial program 89.2%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 81.6

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

   5. Simplified 81.6%

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

   6. Taylor expanded in t around inf

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

      1. /-lowering-/.f64 79.1

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

   8. Simplified 79.1%

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

   if -1.21999999999999999e114 < a < 7.2000000000000004e111

   1. Initial program 73.5%

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

      1. +-commutative N/A

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

      2. clear-num N/A

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

      3. associate-*r/ N/A

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

      4. div-inv N/A

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

      5. times-frac N/A

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

      6. flip3-- N/A

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

      7. clear-num N/A

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

      8. clear-num N/A

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

      9. flip3-- N/A

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

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

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

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

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

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

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

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

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

      14. --lowering--.f64 76.9

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

   4. Applied egg-rr 76.9%

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

   5. Taylor expanded in z around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. mul-1-neg N/A

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

      5. unsub-neg N/A

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

      6. --lowering--.f64 N/A

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

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

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

      8. associate-/l* N/A

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

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

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

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

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

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

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

      12. --lowering--.f64 74.3

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

   7. Simplified 74.3%

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

   8. Taylor expanded in y around inf

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

      1. /-lowering-/.f64 69.8

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

   10. Simplified 69.8%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;a \leq -1.22 \cdot 10^{+114}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \mathbf{elif}\;a \leq 7.2 \cdot 10^{+111}:\\ \;\;\;\;t + \frac{y}{z} \cdot \left(x - t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(y - z, \frac{t}{a}, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 64.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{if}\;z \leq -1.24 \cdot 10^{+120}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 1.55 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(y, \frac{t - x}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ x z) y t)))
   (if (<= z -1.24e+120)
     t_1
     (if (<= z 1.55e+86) (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 <= -1.24e+120) {
		tmp = t_1;
	} else if (z <= 1.55e+86) {
		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 <= -1.24e+120)
		tmp = t_1;
	elseif (z <= 1.55e+86)
		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, -1.24e+120], t$95$1, If[LessEqual[z, 1.55e+86], N[(y * N[(N[(t - x), $MachinePrecision] / a), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.23999999999999998e120 or 1.5500000000000001e86 < z

   1. Initial program 59.5%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

   5. Simplified 83.2%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 75.8

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

   8. Simplified 75.8%

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

   9. Taylor expanded in y around inf

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

   11. Simplified 68.0%

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

   if -1.23999999999999998e120 < z < 1.5500000000000001e86

   1. Initial program 92.2%

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

   3. Taylor expanded in z around 0

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

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

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

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

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

      5. --lowering--.f64 67.2

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

   5. Simplified 67.2%

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

Alternative 11: 54.8% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - z \cdot \frac{t}{a}\\ \mathbf{if}\;a \leq -2.7 \cdot 10^{+98}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 6.3 \cdot 10^{+111}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* z (/ t a)))))
   (if (<= a -2.7e+98) t_1 (if (<= a 6.3e+111) (fma (/ x z) y t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (z * (t / a));
	double tmp;
	if (a <= -2.7e+98) {
		tmp = t_1;
	} else if (a <= 6.3e+111) {
		tmp = fma((x / z), y, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(z * Float64(t / a)))
	tmp = 0.0
	if (a <= -2.7e+98)
		tmp = t_1;
	elseif (a <= 6.3e+111)
		tmp = fma(Float64(x / z), y, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(z * N[(t / a), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[a, -2.7e+98], t$95$1, If[LessEqual[a, 6.3e+111], N[(N[(x / z), $MachinePrecision] * y + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -2.7e98 or 6.3000000000000001e111 < a

   1. Initial program 88.5%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 80.0

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

   5. Simplified 80.0%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 61.2

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

   8. Simplified 61.2%

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

   9. Taylor expanded in t around inf

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

       1. *-commutative N/A

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

       2. associate-/l* N/A

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

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

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

       4. /-lowering-/.f64 69.3

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

   11. Simplified 69.3%

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

   if -2.7e98 < a < 6.3000000000000001e111

   1. Initial program 73.6%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

   5. Simplified 74.0%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 62.1

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

   8. Simplified 62.1%

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

   9. Taylor expanded in y around inf

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

   11. Simplified 57.4%

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

Alternative 12: 52.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{if}\;a \leq -1.35 \cdot 10^{+114}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;a \leq 7.3 \cdot 10^{+112}:\\ \;\;\;\;\mathsf{fma}\left(\frac{x}{z}, y, t\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma x (/ z a) x)))
   (if (<= a -1.35e+114) t_1 (if (<= a 7.3e+112) (fma (/ x z) y t) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(x, (z / a), x);
	double tmp;
	if (a <= -1.35e+114) {
		tmp = t_1;
	} else if (a <= 7.3e+112) {
		tmp = fma((x / z), y, t);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(x, Float64(z / a), x)
	tmp = 0.0
	if (a <= -1.35e+114)
		tmp = t_1;
	elseif (a <= 7.3e+112)
		tmp = fma(Float64(x / z), y, t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x * N[(z / a), $MachinePrecision] + x), $MachinePrecision]}, If[LessEqual[a, -1.35e+114], t$95$1, If[LessEqual[a, 7.3e+112], N[(N[(x / z), $MachinePrecision] * y + t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if a < -1.35e114 or 7.3e112 < a

   1. Initial program 89.2%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 81.6

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

   5. Simplified 81.6%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 62.1

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

   8. Simplified 62.1%

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

   9. Taylor expanded in x around inf

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

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

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

       2. *-lft-identity N/A

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

       3. mul-1-neg N/A

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

       4. cancel-sign-sub N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. +-commutative N/A

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

       8. associate-/l* N/A

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

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

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

       10. /-lowering-/.f64 59.8

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

   11. Simplified 59.8%

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

   if -1.35e114 < a < 7.3e112

   1. Initial program 73.5%

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

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

      6. div-sub N/A

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

      7. associate-/l* N/A

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

      8. associate-/l* N/A

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

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

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

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

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

      11. mul-1-neg N/A

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

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

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

   5. Simplified 73.3%

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

   6. Taylor expanded in x around inf

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

      1. /-lowering-/.f64 61.2

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

   8. Simplified 61.2%

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

   9. Taylor expanded in y around inf

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

   11. Simplified 56.6%

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

Alternative 13: 39.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.6 \cdot 10^{+118}:\\ \;\;\;\;t\\ \mathbf{elif}\;z \leq 2 \cdot 10^{+86}:\\ \;\;\;\;\mathsf{fma}\left(x, \frac{z}{a}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.6e+118) t (if (<= z 2e+86) (fma x (/ z a) x) t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.6e+118) {
		tmp = t;
	} else if (z <= 2e+86) {
		tmp = fma(x, (z / a), x);
	} else {
		tmp = t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.6e+118)
		tmp = t;
	elseif (z <= 2e+86)
		tmp = fma(x, Float64(z / a), x);
	else
		tmp = t;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.6e+118], t, If[LessEqual[z, 2e+86], N[(x * N[(z / a), $MachinePrecision] + x), $MachinePrecision], t]]

Derivation

1. Split input into 2 regimes

2. if z < -3.6e118 or 2e86 < z

   1. Initial program 59.5%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 53.9%

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

   if -3.6e118 < z < 2e86

   1. Initial program 92.2%

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

   3. Taylor expanded in a around inf

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. associate-/l* N/A

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

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

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

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

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

      6. /-lowering-/.f64 N/A

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

      7. --lowering--.f64 71.1

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

   5. Simplified 71.1%

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

   6. Taylor expanded in y around 0

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

      1. mul-1-neg N/A

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

      2. unsub-neg N/A

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

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

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

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

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

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

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

      6. --lowering--.f64 45.0

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

   8. Simplified 45.0%

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

   9. Taylor expanded in x around inf

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

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

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

       2. *-lft-identity N/A

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

       3. mul-1-neg N/A

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

       4. cancel-sign-sub N/A

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

       5. associate-*l/ N/A

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

       6. *-commutative N/A

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

       7. +-commutative N/A

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

       8. associate-/l* N/A

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

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

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

       10. /-lowering-/.f64 42.1

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

   11. Simplified 42.1%

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

Alternative 14: 37.6% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;a \leq -4.6 \cdot 10^{+114}:\\ \;\;\;\;x\\ \mathbf{elif}\;a \leq 6.9 \cdot 10^{+112}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= a -4.6e+114) x (if (<= a 6.9e+112) t x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (a <= -4.6e+114) {
		tmp = x;
	} else if (a <= 6.9e+112) {
		tmp = 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 <= (-4.6d+114)) then
        tmp = x
    else if (a <= 6.9d+112) then
        tmp = 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 <= -4.6e+114) {
		tmp = x;
	} else if (a <= 6.9e+112) {
		tmp = t;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if a <= -4.6e+114:
		tmp = x
	elif a <= 6.9e+112:
		tmp = t
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (a <= -4.6e+114)
		tmp = x;
	elseif (a <= 6.9e+112)
		tmp = t;
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (a <= -4.6e+114)
		tmp = x;
	elseif (a <= 6.9e+112)
		tmp = t;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[a, -4.6e+114], x, If[LessEqual[a, 6.9e+112], t, x]]

Derivation

1. Split input into 2 regimes

2. if a < -4.6000000000000001e114 or 6.8999999999999999e112 < a

   1. Initial program 89.2%

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

   3. Taylor expanded in a around inf

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

   5. Simplified 59.6%

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

   if -4.6000000000000001e114 < a < 6.8999999999999999e112

   1. Initial program 73.5%

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

   3. Taylor expanded in z around inf

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

   5. Simplified 37.8%

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

Alternative 15: 25.2% 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 78.8%

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

3. Taylor expanded in z around inf

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

5. Simplified 29.2%

   \[\leadsto \color{blue}{t} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2024195 
(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)))))