Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1

Percentage Accurate: 97.4% → 96.9%

Time: 9.8s

Alternatives: 17

Speedup: 0.5×

Specification

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \cdot t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 97.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \frac{x - y}{z - y} \cdot t \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 96.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 23000000000000:\\ \;\;\;\;\frac{t\_m \cdot \left(x - y\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (*
  t_s
  (if (<= t_m 23000000000000.0)
    (/ (* t_m (- x y)) (- z y))
    (* (- x y) (/ t_m (- z y))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 23000000000000.0) {
		tmp = (t_m * (x - y)) / (z - y);
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: tmp
    if (t_m <= 23000000000000.0d0) then
        tmp = (t_m * (x - y)) / (z - y)
    else
        tmp = (x - y) * (t_m / (z - y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double tmp;
	if (t_m <= 23000000000000.0) {
		tmp = (t_m * (x - y)) / (z - y);
	} else {
		tmp = (x - y) * (t_m / (z - y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	tmp = 0
	if t_m <= 23000000000000.0:
		tmp = (t_m * (x - y)) / (z - y)
	else:
		tmp = (x - y) * (t_m / (z - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	tmp = 0.0
	if (t_m <= 23000000000000.0)
		tmp = Float64(Float64(t_m * Float64(x - y)) / Float64(z - y));
	else
		tmp = Float64(Float64(x - y) * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	tmp = 0.0;
	if (t_m <= 23000000000000.0)
		tmp = (t_m * (x - y)) / (z - y);
	else
		tmp = (x - y) * (t_m / (z - y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * If[LessEqual[t$95$m, 23000000000000.0], N[(N[(t$95$m * N[(x - y), $MachinePrecision]), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 2.3e13

   1. Initial program 95.9%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

      6. clear-num N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. clear-num N/A

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

      10. flip3-- N/A

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

      11. lift--.f64 N/A

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

      12. div-inv N/A

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

      13. associate-/l* N/A

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

      14. lower-/.f64 N/A

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

      15. lower-*.f64 89.1

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

   4. Applied rewrites 89.1%

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

   if 2.3e13 < t

   1. Initial program 98.2%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

      6. clear-num N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. clear-num N/A

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

      10. flip3-- N/A

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

      11. lift--.f64 N/A

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

      12. div-inv N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 99.7

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

   4. Applied rewrites 99.7%

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

4. Final simplification 91.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 23000000000000:\\ \;\;\;\;\frac{t \cdot \left(x - y\right)}{z - y}\\ \mathbf{else}:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 96.1% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{x}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -50:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (/ x (- z y)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -50.0)
      t_2
      (if (<= t_3 0.4)
        (* t_m (/ (- x y) z))
        (if (<= t_3 2.0)
          (fma t_m (/ z y) t_m)
          (if (<= t_3 2e+192) t_2 (* x (/ t_m (- z y))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (x / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -50.0) {
		tmp = t_2;
	} else if (t_3 <= 0.4) {
		tmp = t_m * ((x - y) / z);
	} else if (t_3 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else if (t_3 <= 2e+192) {
		tmp = t_2;
	} else {
		tmp = x * (t_m / (z - y));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m * Float64(x / Float64(z - y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -50.0)
		tmp = t_2;
	elseif (t_3 <= 0.4)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (t_3 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	elseif (t_3 <= 2e+192)
		tmp = t_2;
	else
		tmp = Float64(x * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -50.0], t$95$2, If[LessEqual[t$95$3, 0.4], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 2e+192], t$95$2, N[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -50 or 2 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.00000000000000008e192

   1. Initial program 98.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 95.4

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

   5. Applied rewrites 95.4%

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

   if -50 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 2.00000000000000008e192 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 80.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. lift--.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{t}{z - y} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -50:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 3: 93.8% accurate, 0.2× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := t\_m \cdot \frac{x}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -0.0002:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t\_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* t_m (/ x (- z y)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -0.0002)
      t_2
      (if (<= t_3 0.4)
        (* (- x y) (/ t_m z))
        (if (<= t_3 2.0)
          (fma t_m (/ z y) t_m)
          (if (<= t_3 2e+192) t_2 (* x (/ t_m (- z y))))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m * (x / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -0.0002) {
		tmp = t_2;
	} else if (t_3 <= 0.4) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else if (t_3 <= 2e+192) {
		tmp = t_2;
	} else {
		tmp = x * (t_m / (z - y));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m * Float64(x / Float64(z - y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -0.0002)
		tmp = t_2;
	elseif (t_3 <= 0.4)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	elseif (t_3 <= 2e+192)
		tmp = t_2;
	else
		tmp = Float64(x * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -0.0002], t$95$2, If[LessEqual[t$95$3, 0.4], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], If[LessEqual[t$95$3, 2e+192], t$95$2, N[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -2.0000000000000001e-4 or 2 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2.00000000000000008e192

   1. Initial program 98.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 95.5

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

   5. Applied rewrites 95.5%

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

   if -2.0000000000000001e-4 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 92.5

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

   5. Applied rewrites 92.5%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 2.00000000000000008e192 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 80.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. lift--.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

      \[\leadsto \color{blue}{\frac{t}{z - y} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 96.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -0.0002:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 94.9% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -50:\\ \;\;\;\;t\_m \cdot \frac{x}{z - y}\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_2 \leq 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z - x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -50.0)
      (* t_m (/ x (- z y)))
      (if (<= t_2 0.4)
        (* t_m (/ (- x y) z))
        (if (<= t_2 1e+25)
          (fma t_m (/ (- z x) y) t_m)
          (* x (/ t_m (- z y)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -50.0) {
		tmp = t_m * (x / (z - y));
	} else if (t_2 <= 0.4) {
		tmp = t_m * ((x - y) / z);
	} else if (t_2 <= 1e+25) {
		tmp = fma(t_m, ((z - x) / y), t_m);
	} else {
		tmp = x * (t_m / (z - y));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -50.0)
		tmp = Float64(t_m * Float64(x / Float64(z - y)));
	elseif (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (t_2 <= 1e+25)
		tmp = fma(t_m, Float64(Float64(z - x) / y), t_m);
	else
		tmp = Float64(x * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -50.0], N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+25], N[(t$95$m * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -50

   1. Initial program 96.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if -50 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000009e25

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

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

   if 1.00000000000000009e25 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 90.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 90.1

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

   5. Applied rewrites 90.1%

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

      1. lift--.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 94.0

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

   7. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\frac{t}{z - y} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -50:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 94.6% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq -50:\\ \;\;\;\;t\_m \cdot \frac{x}{z - y}\\ \mathbf{elif}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x - y}{z}\\ \mathbf{elif}\;t\_2 \leq 10^{+25}:\\ \;\;\;\;t\_m \cdot \frac{y - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 -50.0)
      (* t_m (/ x (- z y)))
      (if (<= t_2 0.4)
        (* t_m (/ (- x y) z))
        (if (<= t_2 1e+25) (* t_m (/ (- y x) y)) (* x (/ t_m (- z y)))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -50.0) {
		tmp = t_m * (x / (z - y));
	} else if (t_2 <= 0.4) {
		tmp = t_m * ((x - y) / z);
	} else if (t_2 <= 1e+25) {
		tmp = t_m * ((y - x) / y);
	} else {
		tmp = x * (t_m / (z - y));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= (-50.0d0)) then
        tmp = t_m * (x / (z - y))
    else if (t_2 <= 0.4d0) then
        tmp = t_m * ((x - y) / z)
    else if (t_2 <= 1d+25) then
        tmp = t_m * ((y - x) / y)
    else
        tmp = x * (t_m / (z - y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= -50.0) {
		tmp = t_m * (x / (z - y));
	} else if (t_2 <= 0.4) {
		tmp = t_m * ((x - y) / z);
	} else if (t_2 <= 1e+25) {
		tmp = t_m * ((y - x) / y);
	} else {
		tmp = x * (t_m / (z - y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= -50.0:
		tmp = t_m * (x / (z - y))
	elif t_2 <= 0.4:
		tmp = t_m * ((x - y) / z)
	elif t_2 <= 1e+25:
		tmp = t_m * ((y - x) / y)
	else:
		tmp = x * (t_m / (z - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= -50.0)
		tmp = Float64(t_m * Float64(x / Float64(z - y)));
	elseif (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(Float64(x - y) / z));
	elseif (t_2 <= 1e+25)
		tmp = Float64(t_m * Float64(Float64(y - x) / y));
	else
		tmp = Float64(x * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= -50.0)
		tmp = t_m * (x / (z - y));
	elseif (t_2 <= 0.4)
		tmp = t_m * ((x - y) / z);
	elseif (t_2 <= 1e+25)
		tmp = t_m * ((y - x) / y);
	else
		tmp = x * (t_m / (z - y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, -50.0], N[(t$95$m * N[(x / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(N[(x - y), $MachinePrecision] / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+25], N[(t$95$m * N[(N[(y - x), $MachinePrecision] / y), $MachinePrecision]), $MachinePrecision], N[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -50

   1. Initial program 96.7%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 94.4

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

   5. Applied rewrites 94.4%

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

   if -50 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 93.9

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

   5. Applied rewrites 93.9%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000009e25

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

      6. clear-num N/A

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

      7. *-commutative N/A

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

      8. lower-*.f64 N/A

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

      9. clear-num N/A

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

      10. flip3-- N/A

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

      11. lift--.f64 N/A

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

      12. lower-/.f64 99.8

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

   4. Applied rewrites 99.8%

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

      1. lift--.f64 N/A

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

      2. frac-2neg N/A

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

      3. metadata-eval N/A

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

      4. lift--.f64 N/A

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

      5. associate-*l/ N/A

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

      6. neg-mul-1 N/A

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

      7. lower-/.f64 N/A

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

      8. neg-sub0 N/A

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

      9. lift--.f64 N/A

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

      10. sub-neg N/A

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

      11. lift-neg.f64 N/A

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

      12. +-commutative N/A

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

      13. associate--r+ N/A

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

      14. neg-sub0 N/A

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

      15. lift-neg.f64 N/A

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

      16. remove-double-neg N/A

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

      17. lower--.f64 N/A

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

      18. neg-sub0 N/A

          \[\leadsto \frac{y - x}{\color{blue}{0 - \left(z - y\right)}} \cdot t \]

      19. lift--.f64 N/A

          \[\leadsto \frac{y - x}{0 - \color{blue}{\left(z - y\right)}} \cdot t \]

      20. sub-neg N/A

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

      21. lift-neg.f64 N/A

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

      22. +-commutative N/A

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

      23. associate--r+ N/A

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

      24. neg-sub0 N/A

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

      25. lift-neg.f64 N/A

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

      26. remove-double-neg N/A

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

      27. lower--.f64 100.0

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

   6. Applied rewrites 100.0%

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

   7. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 98.1

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

   9. Applied rewrites 98.1%

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

   if 1.00000000000000009e25 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 90.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 90.1

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

   5. Applied rewrites 90.1%

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

      1. lift--.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 94.0

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

   7. Applied rewrites 94.0%

      \[\leadsto \color{blue}{\frac{t}{z - y} \cdot x} \]
3. Recombined 4 regimes into one program.

4. Final simplification 95.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -50:\\ \;\;\;\;t \cdot \frac{x}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x - y}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{+25}:\\ \;\;\;\;t \cdot \frac{y - x}{y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 91.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \frac{t\_m}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq -50:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_3 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* x (/ t_m (- z y)))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 -50.0)
      t_2
      (if (<= t_3 0.4)
        (* (- x y) (/ t_m z))
        (if (<= t_3 20.0) (fma t_m (/ z y) t_m) t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = x * (t_m / (z - y));
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= -50.0) {
		tmp = t_2;
	} else if (t_3 <= 0.4) {
		tmp = (x - y) * (t_m / z);
	} else if (t_3 <= 20.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(x * Float64(t_m / Float64(z - y)))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= -50.0)
		tmp = t_2;
	elseif (t_3 <= 0.4)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_3 <= 20.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, -50.0], t$95$2, If[LessEqual[t$95$3, 0.4], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$3, 20.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], t$95$2]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < -50 or 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 93.2%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 92.1

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

   5. Applied rewrites 92.1%

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

      1. lift--.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 87.9

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

   7. Applied rewrites 87.9%

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

   if -50 < (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 92.6

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

   5. Applied rewrites 92.6%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.9

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

   8. Applied rewrites 98.9%

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

4. Final simplification 93.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq -50:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 92.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{z - y}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z - x}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ t_m (- z y))) (t_3 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_3 0.4)
      (* (- x y) t_2)
      (if (<= t_3 1e+25) (fma t_m (/ (- z x) y) t_m) (* x t_2))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = t_m / (z - y);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 0.4) {
		tmp = (x - y) * t_2;
	} else if (t_3 <= 1e+25) {
		tmp = fma(t_m, ((z - x) / y), t_m);
	} else {
		tmp = x * t_2;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(t_m / Float64(z - y))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 0.4)
		tmp = Float64(Float64(x - y) * t_2);
	elseif (t_3 <= 1e+25)
		tmp = fma(t_m, Float64(Float64(z - x) / y), t_m);
	else
		tmp = Float64(x * t_2);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 0.4], N[(N[(x - y), $MachinePrecision] * t$95$2), $MachinePrecision], If[LessEqual[t$95$3, 1e+25], N[(t$95$m * N[(N[(z - x), $MachinePrecision] / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(x * t$95$2), $MachinePrecision]]]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing
   3. Step-by-step derivation

      1. lift--.f64 N/A

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

      2. lift--.f64 N/A

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

      3. div-inv N/A

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

      4. lift--.f64 N/A

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

      5. flip3-- N/A

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

      6. clear-num N/A

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

      7. associate-*r* N/A

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

      8. *-commutative N/A

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

      9. clear-num N/A

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

      10. flip3-- N/A

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

      11. lift--.f64 N/A

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

      12. div-inv N/A

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

      13. *-commutative N/A

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

      14. lower-*.f64 N/A

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

      15. lower-/.f64 92.0

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

   4. Applied rewrites 92.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000009e25

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

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

      1. associate--l+ N/A

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

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

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

      3. div-sub N/A

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

      4. +-commutative N/A

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

      5. mul-1-neg N/A

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

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

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

      7. associate-/l* N/A

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

      8. distribute-rgt-neg-in N/A

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

      9. mul-1-neg N/A

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

      10. lower-fma.f64 N/A

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

   5. Applied rewrites 99.0%

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

   if 1.00000000000000009e25 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 90.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 90.1

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

   5. Applied rewrites 90.1%

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

      1. lift--.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 94.0

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

   7. Applied rewrites 94.0%

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

4. Final simplification 95.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t}{z - y}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z - x}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 78.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;\left(x - y\right) \cdot \frac{t\_m}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-t\_m\right)}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* (- x y) (/ t_m z))
      (if (<= t_2 2.0) (fma t_m (/ z y) t_m) (/ (* x (- t_m)) y))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = (x - y) * (t_m / z);
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (x * -t_m) / y;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(Float64(x - y) * Float64(t_m / z));
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(x * Float64(-t_m)) / y);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(N[(x - y), $MachinePrecision] * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(x * (-t$95$m)), $MachinePrecision] / y), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in z around inf

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

      1. *-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-*.f64 N/A

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

      4. lower--.f64 N/A

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

      5. lower-/.f64 80.0

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

   5. Applied rewrites 80.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 59.8

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

   8. Applied rewrites 59.8%

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

Alternative 9: 70.5% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-t\_m\right)}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* t_m (/ x z))
      (if (<= t_2 2.0) (fma t_m (/ z y) t_m) (/ (* x (- t_m)) y))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (x * -t_m) / y;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(x * Float64(-t_m)) / y);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(x * (-t$95$m)), $MachinePrecision] / y), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. *-commutative N/A

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

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

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

      6. neg-mul-1 N/A

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

      7. lower-*.f64 N/A

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

      8. neg-mul-1 N/A

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

      9. lower-neg.f64 59.8

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

   8. Applied rewrites 59.8%

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

4. Final simplification 75.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{x \cdot \left(-t\right)}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 71.0% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;t\_m \cdot \frac{x}{-y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* t_m (/ x z))
      (if (<= t_2 2.0) (fma t_m (/ z y) t_m) (* t_m (/ x (- y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 2.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = t_m * (x / -y);
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 2.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(t_m * Float64(x / Float64(-y)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 2.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(t$95$m * N[(x / (-y)), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 2

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 99.8

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

   5. Applied rewrites 99.8%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 99.8

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

   8. Applied rewrites 99.8%

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

   if 2 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 89.6

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

   5. Applied rewrites 89.6%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 59.6

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

   8. Applied rewrites 59.6%

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

4. Final simplification 75.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 2:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;t \cdot \frac{x}{-y}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 70.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \frac{t\_m}{y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* t_m (/ x z))
      (if (<= t_2 20.0) (fma t_m (/ z y) t_m) (- (* x (/ t_m y))))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 20.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = -(x * (t_m / y));
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 20.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(-Float64(x * Float64(t_m / y)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], (-N[(x * N[(t$95$m / y), $MachinePrecision]), $MachinePrecision])]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 90.7

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

   5. Applied rewrites 90.7%

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

   6. Taylor expanded in z around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 60.2

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

   8. Applied rewrites 60.2%

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

      1. lift-neg.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. *-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. clear-num N/A

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

      6. associate-/r/ N/A

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

      7. associate-*r* N/A

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

      8. div-inv N/A

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

      9. lower-*.f64 N/A

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

      10. lower-/.f64 57.1

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

   10. Applied rewrites 57.1%

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

4. Final simplification 74.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;-x \cdot \frac{t}{y}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 70.4% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t\_m, \frac{z}{y}, t\_m\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* t_m (/ x z))
      (if (<= t_2 20.0) (fma t_m (/ z y) t_m) (/ (* t_m x) z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 20.0) {
		tmp = fma(t_m, (z / y), t_m);
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 20.0)
		tmp = fma(t_m, Float64(z / y), t_m);
	else
		tmp = Float64(Float64(t_m * x) / z);
	end
	return Float64(t_s * tmp)
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], N[(t$95$m * N[(z / y), $MachinePrecision] + t$95$m), $MachinePrecision], N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around 0

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 98.9

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

   5. Applied rewrites 98.9%

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

   6. Taylor expanded in y around inf

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

      1. +-commutative N/A

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

      2. associate-/l* N/A

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

      3. lower-fma.f64 N/A

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

      4. lower-/.f64 98.9

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

   8. Applied rewrites 98.9%

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

   if 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;\mathsf{fma}\left(t, \frac{z}{y}, t\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 13: 70.1% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 20:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;\frac{t\_m \cdot x}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4) (* t_m (/ x z)) (if (<= t_2 20.0) t_m (/ (* t_m x) z))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 20.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 0.4d0) then
        tmp = t_m * (x / z)
    else if (t_2 <= 20.0d0) then
        tmp = t_m
    else
        tmp = (t_m * x) / z
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 20.0) {
		tmp = t_m;
	} else {
		tmp = (t_m * x) / z;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 0.4:
		tmp = t_m * (x / z)
	elif t_2 <= 20.0:
		tmp = t_m
	else:
		tmp = (t_m * x) / z
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 20.0)
		tmp = t_m;
	else
		tmp = Float64(Float64(t_m * x) / z);
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 0.4)
		tmp = t_m * (x / z);
	elseif (t_2 <= 20.0)
		tmp = t_m;
	else
		tmp = (t_m * x) / z;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 20.0], t$95$m, N[(N[(t$95$m * x), $MachinePrecision] / z), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 20

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot t \]
   4. Step-by-step derivation

   5. Applied rewrites 98.2%

      \[\leadsto \color{blue}{1} \cdot t \]
   6. Step-by-step derivation

      1. *-lft-identity 98.2

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

   7. Applied rewrites 98.2%

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

   if 20 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 91.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 48.0

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

   5. Applied rewrites 48.0%

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

4. Final simplification 72.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 20:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot x}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 69.6% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 0.4:\\ \;\;\;\;t\_m \cdot \frac{x}{z}\\ \mathbf{elif}\;t\_2 \leq 10^{+25}:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t\_m}{z}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (*
    t_s
    (if (<= t_2 0.4)
      (* t_m (/ x z))
      (if (<= t_2 1e+25) t_m (* x (/ t_m z)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 1e+25) {
		tmp = t_m;
	} else {
		tmp = x * (t_m / z);
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 0.4d0) then
        tmp = t_m * (x / z)
    else if (t_2 <= 1d+25) then
        tmp = t_m
    else
        tmp = x * (t_m / z)
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 0.4) {
		tmp = t_m * (x / z);
	} else if (t_2 <= 1e+25) {
		tmp = t_m;
	} else {
		tmp = x * (t_m / z);
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 0.4:
		tmp = t_m * (x / z)
	elif t_2 <= 1e+25:
		tmp = t_m
	else:
		tmp = x * (t_m / z)
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 0.4)
		tmp = Float64(t_m * Float64(x / z));
	elseif (t_2 <= 1e+25)
		tmp = t_m;
	else
		tmp = Float64(x * Float64(t_m / z));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 0.4)
		tmp = t_m * (x / z);
	elseif (t_2 <= 1e+25)
		tmp = t_m;
	else
		tmp = x * (t_m / z);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 0.4], N[(t$95$m * N[(x / z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 1e+25], t$95$m, N[(x * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002

   1. Initial program 96.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 64.0

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

   5. Applied rewrites 64.0%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000009e25

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot t \]
   4. Step-by-step derivation

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{1} \cdot t \]
   6. Step-by-step derivation

      1. *-lft-identity 92.8

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

   7. Applied rewrites 92.8%

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

   if 1.00000000000000009e25 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 90.1%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 46.4

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

   5. Applied rewrites 46.4%

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 50.3

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

   7. Applied rewrites 50.3%

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

4. Final simplification 72.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;t \cdot \frac{x}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{+25}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 67.9% accurate, 0.4× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := x \cdot \frac{t\_m}{z}\\ t_3 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_3 \leq 0.4:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq 10^{+25}:\\ \;\;\;\;t\_m\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (* x (/ t_m z))) (t_3 (/ (- x y) (- z y))))
   (* t_s (if (<= t_3 0.4) t_2 (if (<= t_3 1e+25) t_m t_2)))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = x * (t_m / z);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 0.4) {
		tmp = t_2;
	} else if (t_3 <= 1e+25) {
		tmp = t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: t_3
    real(8) :: tmp
    t_2 = x * (t_m / z)
    t_3 = (x - y) / (z - y)
    if (t_3 <= 0.4d0) then
        tmp = t_2
    else if (t_3 <= 1d+25) then
        tmp = t_m
    else
        tmp = t_2
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = x * (t_m / z);
	double t_3 = (x - y) / (z - y);
	double tmp;
	if (t_3 <= 0.4) {
		tmp = t_2;
	} else if (t_3 <= 1e+25) {
		tmp = t_m;
	} else {
		tmp = t_2;
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = x * (t_m / z)
	t_3 = (x - y) / (z - y)
	tmp = 0
	if t_3 <= 0.4:
		tmp = t_2
	elif t_3 <= 1e+25:
		tmp = t_m
	else:
		tmp = t_2
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(x * Float64(t_m / z))
	t_3 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_3 <= 0.4)
		tmp = t_2;
	elseif (t_3 <= 1e+25)
		tmp = t_m;
	else
		tmp = t_2;
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = x * (t_m / z);
	t_3 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_3 <= 0.4)
		tmp = t_2;
	elseif (t_3 <= 1e+25)
		tmp = t_m;
	else
		tmp = t_2;
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(x * N[(t$95$m / z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$3, 0.4], t$95$2, If[LessEqual[t$95$3, 1e+25], t$95$m, t$95$2]]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 0.40000000000000002 or 1.00000000000000009e25 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 94.3%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around 0

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

      1. lower-/.f64 58.5

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

   5. Applied rewrites 58.5%

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

      1. lift-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lift-/.f64 N/A

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

      4. div-inv N/A

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

      5. *-commutative N/A

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

      6. associate-*r* N/A

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

      7. div-inv N/A

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

      8. lower-*.f64 N/A

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

      9. lower-/.f64 56.8

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

   7. Applied rewrites 56.8%

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

   if 0.40000000000000002 < (/.f64 (-.f64 x y) (-.f64 z y)) < 1.00000000000000009e25

   1. Initial program 100.0%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in y around inf

      \[\leadsto \color{blue}{1} \cdot t \]
   4. Step-by-step derivation

   5. Applied rewrites 92.8%

      \[\leadsto \color{blue}{1} \cdot t \]
   6. Step-by-step derivation

      1. *-lft-identity 92.8

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

   7. Applied rewrites 92.8%

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

4. Final simplification 70.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 0.4:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \mathbf{elif}\;\frac{x - y}{z - y} \leq 10^{+25}:\\ \;\;\;\;t\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 98.0% accurate, 0.5× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{x - y}{z - y}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_2 \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t\_m \cdot t\_2\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t\_m}{z - y}\\ \end{array} \end{array} \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m)
 :precision binary64
 (let* ((t_2 (/ (- x y) (- z y))))
   (* t_s (if (<= t_2 2e+192) (* t_m t_2) (* x (/ t_m (- z y)))))))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 2e+192) {
		tmp = t_m * t_2;
	} else {
		tmp = x * (t_m / (z - y));
	}
	return t_s * tmp;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    real(8) :: t_2
    real(8) :: tmp
    t_2 = (x - y) / (z - y)
    if (t_2 <= 2d+192) then
        tmp = t_m * t_2
    else
        tmp = x * (t_m / (z - y))
    end if
    code = t_s * tmp
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	double t_2 = (x - y) / (z - y);
	double tmp;
	if (t_2 <= 2e+192) {
		tmp = t_m * t_2;
	} else {
		tmp = x * (t_m / (z - y));
	}
	return t_s * tmp;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	t_2 = (x - y) / (z - y)
	tmp = 0
	if t_2 <= 2e+192:
		tmp = t_m * t_2
	else:
		tmp = x * (t_m / (z - y))
	return t_s * tmp

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	t_2 = Float64(Float64(x - y) / Float64(z - y))
	tmp = 0.0
	if (t_2 <= 2e+192)
		tmp = Float64(t_m * t_2);
	else
		tmp = Float64(x * Float64(t_m / Float64(z - y)));
	end
	return Float64(t_s * tmp)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, x, y, z, t_m)
	t_2 = (x - y) / (z - y);
	tmp = 0.0;
	if (t_2 <= 2e+192)
		tmp = t_m * t_2;
	else
		tmp = x * (t_m / (z - y));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := Block[{t$95$2 = N[(N[(x - y), $MachinePrecision] / N[(z - y), $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$2, 2e+192], N[(t$95$m * t$95$2), $MachinePrecision], N[(x * N[(t$95$m / N[(z - y), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 x y) (-.f64 z y)) < 2.00000000000000008e192

   1. Initial program 98.2%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   if 2.00000000000000008e192 < (/.f64 (-.f64 x y) (-.f64 z y))

   1. Initial program 80.8%

      \[\frac{x - y}{z - y} \cdot t \]
   2. Add Preprocessing

   3. Taylor expanded in x around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 80.8

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

   5. Applied rewrites 80.8%

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

      1. lift--.f64 N/A

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

      2. frac-2neg N/A

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

      3. frac-2neg N/A

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

      4. lift-/.f64 N/A

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

      5. *-commutative N/A

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

      6. lift-/.f64 N/A

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

      7. clear-num N/A

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

      8. associate-/r/ N/A

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

      9. lift-/.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift-/.f64 N/A

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

      12. div-inv N/A

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

      13. lower-*.f64 N/A

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

      14. lower-/.f64 99.8

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

   7. Applied rewrites 99.8%

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

4. Final simplification 98.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\frac{x - y}{z - y} \leq 2 \cdot 10^{+192}:\\ \;\;\;\;t \cdot \frac{x - y}{z - y}\\ \mathbf{else}:\\ \;\;\;\;x \cdot \frac{t}{z - y}\\ \end{array} \]
5. Add Preprocessing

Alternative 17: 35.8% accurate, 23.0× speedup

Math

\[\begin{array}{l} t\_m = \left|t\right| \\ t\_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot t\_m \end{array} \]

FPCore

t\_m = (fabs.f64 t)
t\_s = (copysign.f64 #s(literal 1 binary64) t)
(FPCore (t_s x y z t_m) :precision binary64 (* t_s t_m))

C

t\_m = fabs(t);
t\_s = copysign(1.0, t);
double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * t_m;
}

Fortran

t\_m = abs(t)
t\_s = copysign(1.0d0, t)
real(8) function code(t_s, x, y, z, t_m)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t_m
    code = t_s * t_m
end function

Java

t\_m = Math.abs(t);
t\_s = Math.copySign(1.0, t);
public static double code(double t_s, double x, double y, double z, double t_m) {
	return t_s * t_m;
}

Python

t\_m = math.fabs(t)
t\_s = math.copysign(1.0, t)
def code(t_s, x, y, z, t_m):
	return t_s * t_m

Julia

t\_m = abs(t)
t\_s = copysign(1.0, t)
function code(t_s, x, y, z, t_m)
	return Float64(t_s * t_m)
end

MATLAB

t\_m = abs(t);
t\_s = sign(t) * abs(1.0);
function tmp = code(t_s, x, y, z, t_m)
	tmp = t_s * t_m;
end

Wolfram

t\_m = N[Abs[t], $MachinePrecision]
t\_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, x_, y_, z_, t$95$m_] := N[(t$95$s * t$95$m), $MachinePrecision]

Derivation

1. Initial program 96.5%

   \[\frac{x - y}{z - y} \cdot t \]
2. Add Preprocessing

3. Taylor expanded in y around inf

   \[\leadsto \color{blue}{1} \cdot t \]
4. Step-by-step derivation

5. Applied rewrites 38.1%

   \[\leadsto \color{blue}{1} \cdot t \]
6. Step-by-step derivation

   1. *-lft-identity 38.1

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

7. Applied rewrites 38.1%

   \[\leadsto \color{blue}{t} \]
8. Add Preprocessing

Developer Target 1: 97.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \frac{t}{\frac{z - y}{x - y}} \end{array} \]

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Reproduce

herbie shell --seed 2024219 
(FPCore (x y z t)
  :name "Numeric.Signal.Multichannel:$cput from hsignal-0.2.7.1"
  :precision binary64

  :alt
  (! :herbie-platform default (/ t (/ (- z y) (- x y))))

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