Numeric.Signal:interpolate from hsignal-0.2.7.1

Percentage Accurate: 80.7% → 95.3%

Time: 12.3s

Alternatives: 19

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: 80.7% 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: 95.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{x - t}{a - z} \cdot \left(y - z\right)\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{-1}{\frac{z - a}{z - y}}, x\right)\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{-1}{z - a} \cdot \left(z - y\right), x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (/ (- x t) (- a z)) (- y z)))))
   (if (<= t_1 -5e-304)
     (fma (- x t) (/ -1.0 (/ (- z a) (- z y))) x)
     (if (<= t_1 0.0)
       (fma (/ (- t x) z) (- a y) t)
       (fma (- x t) (* (/ -1.0 (- z a)) (- z y)) x)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (((x - t) / (a - z)) * (y - z));
	double tmp;
	if (t_1 <= -5e-304) {
		tmp = fma((x - t), (-1.0 / ((z - a) / (z - y))), x);
	} else if (t_1 <= 0.0) {
		tmp = fma(((t - x) / z), (a - y), t);
	} else {
		tmp = fma((x - t), ((-1.0 / (z - a)) * (z - y)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(Float64(x - t) / Float64(a - z)) * Float64(y - z)))
	tmp = 0.0
	if (t_1 <= -5e-304)
		tmp = fma(Float64(x - t), Float64(-1.0 / Float64(Float64(z - a) / Float64(z - y))), x);
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(Float64(t - x) / z), Float64(a - y), t);
	else
		tmp = fma(Float64(x - t), Float64(Float64(-1.0 / Float64(z - a)) * Float64(z - y)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(N[(x - t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e-304], N[(N[(x - t), $MachinePrecision] * N[(-1.0 / N[(N[(z - a), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t$95$1, 0.0], N[(N[(N[(t - x), $MachinePrecision] / z), $MachinePrecision] * N[(a - y), $MachinePrecision] + t), $MachinePrecision], N[(N[(x - t), $MachinePrecision] * N[(N[(-1.0 / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -4.99999999999999965e-304

   1. Initial program 92.5%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(t - x\right)}, \frac{1}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(y - z\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(t - x\right)}, \frac{1}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(y - z\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)}, \frac{1}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(y - z\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + t\right)}, \frac{1}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(y - z\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - t}, \frac{1}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(y - z\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. frac-2neg N/A

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

      20. metadata-eval N/A

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

      21. remove-double-neg N/A

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

      22. lower-/.f64 96.9

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

   4. Applied rewrites 96.9%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. clear-num N/A

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

      5. metadata-eval N/A

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

      6. lower-/.f64 N/A

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

      7. metadata-eval N/A

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

      8. lower-/.f64 N/A

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

      9. mul-1-neg N/A

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

      10. lower-neg.f64 97.0

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

   6. Applied rewrites 97.0%

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

   if -4.99999999999999965e-304 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 3.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 99.8%

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

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

   1. Initial program 92.0%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. frac-2neg N/A

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

      7. div-inv N/A

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

      8. associate-*l* N/A

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

      9. lower-fma.f64 N/A

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

      10. neg-sub0 N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{0 - \left(t - x\right)}, \frac{1}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(y - z\right), x\right) \]

      11. lift--.f64 N/A

          \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(t - x\right)}, \frac{1}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(y - z\right), x\right) \]

      12. sub-neg N/A

          \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(t + \left(\mathsf{neg}\left(x\right)\right)\right)}, \frac{1}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(y - z\right), x\right) \]

      13. +-commutative N/A

          \[\leadsto \mathsf{fma}\left(0 - \color{blue}{\left(\left(\mathsf{neg}\left(x\right)\right) + t\right)}, \frac{1}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(y - z\right), x\right) \]

      14. associate--r+ N/A

          \[\leadsto \mathsf{fma}\left(\color{blue}{\left(0 - \left(\mathsf{neg}\left(x\right)\right)\right) - t}, \frac{1}{\mathsf{neg}\left(\left(a - z\right)\right)} \cdot \left(y - z\right), x\right) \]

      15. neg-sub0 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 N/A

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

      18. lower-*.f64 N/A

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

      19. frac-2neg N/A

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

      20. metadata-eval N/A

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

      21. remove-double-neg N/A

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

      22. lower-/.f64 96.3

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

   4. Applied rewrites 96.3%

      \[\leadsto \color{blue}{\mathsf{fma}\left(x - t, \frac{-1}{a - z} \cdot \left(y - z\right), x\right)} \]
3. Recombined 3 regimes into one program.

4. Final simplification 97.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{x - t}{a - z} \cdot \left(y - z\right) \leq -5 \cdot 10^{-304}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{-1}{\frac{z - a}{z - y}}, x\right)\\ \mathbf{elif}\;x - \frac{x - t}{a - z} \cdot \left(y - z\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - x}{z}, a - y, t\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(x - t, \frac{-1}{z - a} \cdot \left(z - y\right), x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 90.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{x - t}{a - z} \cdot \left(y - z\right)\\ \mathbf{if}\;t\_1 \leq -2 \cdot 10^{-168}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_1 \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{a - y}{z}, t\right)\\ \mathbf{elif}\;t\_1 \leq 2 \cdot 10^{+293}:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - t\right) \cdot y}{z - a} + x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (/ (- x t) (- a z)) (- y z)))))
   (if (<= t_1 -2e-168)
     t_1
     (if (<= t_1 0.0)
       (fma (- t x) (/ (- a y) z) t)
       (if (<= t_1 2e+293) t_1 (+ (/ (* (- x t) y) (- z a)) x))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - (((x - t) / (a - z)) * (y - z));
	double tmp;
	if (t_1 <= -2e-168) {
		tmp = t_1;
	} else if (t_1 <= 0.0) {
		tmp = fma((t - x), ((a - y) / z), t);
	} else if (t_1 <= 2e+293) {
		tmp = t_1;
	} else {
		tmp = (((x - t) * y) / (z - a)) + x;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(Float64(x - t) / Float64(a - z)) * Float64(y - z)))
	tmp = 0.0
	if (t_1 <= -2e-168)
		tmp = t_1;
	elseif (t_1 <= 0.0)
		tmp = fma(Float64(t - x), Float64(Float64(a - y) / z), t);
	elseif (t_1 <= 2e+293)
		tmp = t_1;
	else
		tmp = Float64(Float64(Float64(Float64(x - t) * y) / Float64(z - a)) + x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(N[(x - t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision] * N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -2e-168], t$95$1, If[LessEqual[t$95$1, 0.0], N[(N[(t - x), $MachinePrecision] * N[(N[(a - y), $MachinePrecision] / z), $MachinePrecision] + t), $MachinePrecision], If[LessEqual[t$95$1, 2e+293], t$95$1, N[(N[(N[(N[(x - t), $MachinePrecision] * y), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]]]

Derivation

1. Split input into 3 regimes

2. if (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < -2.0000000000000001e-168 or 0.0 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 1.9999999999999998e293

   1. Initial program 92.6%

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

   if -2.0000000000000001e-168 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z)))) < 0.0

   1. Initial program 14.8%

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

   3. Taylor expanded in z around inf

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

      1. +-commutative N/A

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

      2. associate--l+ N/A

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

      3. sub-neg N/A

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

      4. mul-1-neg N/A

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

      5. remove-double-neg N/A

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

      6. +-commutative N/A

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

      7. associate-+r+ N/A

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

      8. associate-/l* N/A

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

      9. associate-/l* N/A

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

      10. associate-*r* N/A

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

      11. distribute-rgt-out N/A

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

      12. lower-fma.f64 N/A

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

   5. Applied rewrites 85.5%

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

   7. Applied rewrites 84.7%

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

   if 1.9999999999999998e293 < (+.f64 x (*.f64 (-.f64 y z) (/.f64 (-.f64 t x) (-.f64 a z))))

   1. Initial program 86.1%

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

   3. Taylor expanded in z around inf

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

      1. lower--.f64 8.1

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

   5. Applied rewrites 8.1%

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

   6. Taylor expanded in y around inf

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower--.f64 85.6

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

   8. Applied rewrites 85.6%

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

4. Final simplification 90.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;x - \frac{x - t}{a - z} \cdot \left(y - z\right) \leq -2 \cdot 10^{-168}:\\ \;\;\;\;x - \frac{x - t}{a - z} \cdot \left(y - z\right)\\ \mathbf{elif}\;x - \frac{x - t}{a - z} \cdot \left(y - z\right) \leq 0:\\ \;\;\;\;\mathsf{fma}\left(t - x, \frac{a - y}{z}, t\right)\\ \mathbf{elif}\;x - \frac{x - t}{a - z} \cdot \left(y - z\right) \leq 2 \cdot 10^{+293}:\\ \;\;\;\;x - \frac{x - t}{a - z} \cdot \left(y - z\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(x - t\right) \cdot y}{z - a} + x\\ \end{array} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2024230 
(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)))))