Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A

Percentage Accurate: 85.4% → 98.4%

Time: 5.6s

Alternatives: 13

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 85.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.4% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. sub-flip N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-*l/ N/A

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

   8. mult-flip-rev N/A

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

   9. remove-double-neg N/A

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

   10. lower-fma.f64 N/A

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

   11. mult-flip-rev N/A

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

   12. frac-2neg N/A

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

   13. lift--.f64 N/A

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

   14. sub-negate-rev N/A

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

   15. lower-/.f64 N/A

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

   16. lower--.f64 N/A

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

   17. lift--.f64 N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 98.4%

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

3. Applied rewrites 98.4%

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

Alternative 2: 95.5% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 85.4%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. add-flip N/A

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

   4. sub-flip N/A

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

   5. lift-/.f64 N/A

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

   6. lift-*.f64 N/A

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

   7. associate-/l* N/A

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

   8. *-commutative N/A

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

   9. lift--.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. distribute-rgt-neg-out N/A

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

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

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

   13. remove-double-neg N/A

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

   14. lower-fma.f64 N/A

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

   15. distribute-neg-frac2 N/A

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

   16. lower-/.f64 N/A

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

   17. lift--.f64 N/A

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

   18. sub-negate-rev N/A

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

   19. lower--.f64 N/A

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

   20. lower--.f64 95.5%

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

3. Applied rewrites 95.5%

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

Alternative 3: 86.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+39}:\\ \;\;\;\;\frac{t}{\frac{a - z}{y - z}}\\ \mathbf{elif}\;t\_1 \leq 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{t}{z - a} \cdot \left(z - y\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (/ (* (- y z) t) (- a z))))
  (if (<= t_1 -5e+39)
    (/ t (/ (- a z) (- y z)))
    (if (<= t_1 1e+135)
      (fma (/ z (- z a)) t x)
      (* (/ t (- z a)) (- z y))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = ((y - z) * t) / (a - z);
	double tmp;
	if (t_1 <= -5e+39) {
		tmp = t / ((a - z) / (y - z));
	} else if (t_1 <= 1e+135) {
		tmp = fma((z / (z - a)), t, x);
	} else {
		tmp = (t / (z - a)) * (z - y);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(Float64(y - z) * t) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -5e+39)
		tmp = Float64(t / Float64(Float64(a - z) / Float64(y - z)));
	elseif (t_1 <= 1e+135)
		tmp = fma(Float64(z / Float64(z - a)), t, x);
	else
		tmp = Float64(Float64(t / Float64(z - a)) * Float64(z - y));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+39], N[(t / N[(N[(a - z), $MachinePrecision] / N[(y - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$1, 1e+135], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -5.0000000000000002e39

   1. Initial program 85.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. mul-1-neg N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. distribute-rgt-neg-out N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. distribute-neg-frac2 N/A

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

      20. mul-1-neg N/A

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

      21. frac-2neg N/A

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

      22. lower-/.f64 46.1%

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

   6. Applied rewrites 46.1%

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

      1. lift-*.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-/r/ N/A

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

      4. lift-/.f64 N/A

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

      5. lower-/.f64 49.1%

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lift--.f64 N/A

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

      9. sub-negate-rev N/A

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

      10. lift--.f64 N/A

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

      11. sub-negate-rev N/A

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

      12. lower-/.f64 N/A

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

      13. lower--.f64 N/A

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

      14. lower--.f64 49.1%

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

   8. Applied rewrites 49.1%

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

   if -5.0000000000000002e39 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 9.9999999999999996e134

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 72.5%

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

   if 9.9999999999999996e134 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 85.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. mul-1-neg N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. distribute-rgt-neg-out N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. distribute-neg-frac2 N/A

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

      20. mul-1-neg N/A

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

      21. frac-2neg N/A

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

      22. lower-/.f64 46.1%

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

   6. Applied rewrites 46.1%

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

Alternative 4: 84.0% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{t}{z - a} \cdot \left(z - y\right)\\ t_2 := \frac{\left(y - z\right) \cdot t}{a - z}\\ \mathbf{if}\;t\_2 \leq -2 \cdot 10^{+48}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{+135}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (* (/ t (- z a)) (- z y)))
       (t_2 (/ (* (- y z) t) (- a z))))
  (if (<= t_2 -2e+48)
    t_1
    (if (<= t_2 1e+135) (fma (/ z (- z a)) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t / (z - a)) * (z - y);
	double t_2 = ((y - z) * t) / (a - z);
	double tmp;
	if (t_2 <= -2e+48) {
		tmp = t_1;
	} else if (t_2 <= 1e+135) {
		tmp = fma((z / (z - a)), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t / Float64(z - a)) * Float64(z - y))
	t_2 = Float64(Float64(Float64(y - z) * t) / Float64(a - z))
	tmp = 0.0
	if (t_2 <= -2e+48)
		tmp = t_1;
	elseif (t_2 <= 1e+135)
		tmp = fma(Float64(z / Float64(z - a)), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - y), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[(y - z), $MachinePrecision] * t), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -2e+48], t$95$1, If[LessEqual[t$95$2, 1e+135], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < -2.0000000000000001e48 or 9.9999999999999996e134 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z))

   1. Initial program 85.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

      1. lift-/.f64 N/A

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

      2. frac-2neg N/A

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

      3. mul-1-neg N/A

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

      4. lift--.f64 N/A

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

      5. sub-negate-rev N/A

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

      6. lift--.f64 N/A

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

      7. associate-*l/ N/A

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

      8. lift-/.f64 N/A

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

      9. lift-*.f64 N/A

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

      10. associate-*r* N/A

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

      11. lift--.f64 N/A

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

      12. sub-negate-rev N/A

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

      13. lift--.f64 N/A

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

      14. distribute-rgt-neg-out N/A

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

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

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

      16. lower-*.f64 N/A

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

      17. lift-/.f64 N/A

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

      18. associate-*l/ N/A

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

      19. distribute-neg-frac2 N/A

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

      20. mul-1-neg N/A

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

      21. frac-2neg N/A

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

      22. lower-/.f64 46.1%

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

   6. Applied rewrites 46.1%

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

   if -2.0000000000000001e48 < (/.f64 (*.f64 (-.f64 y z) t) (-.f64 a z)) < 9.9999999999999996e134

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 72.5%

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

Alternative 5: 83.8% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -3.9 \cdot 10^{+21}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-11}:\\ \;\;\;\;x + \frac{t \cdot y}{a - z}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, t, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= z -3.9e+21)
  (fma (- 1.0 (/ y z)) t x)
  (if (<= z 5.1e-11)
    (+ x (/ (* t y) (- a z)))
    (fma (/ z (- z a)) t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.9e+21) {
		tmp = fma((1.0 - (y / z)), t, x);
	} else if (z <= 5.1e-11) {
		tmp = x + ((t * y) / (a - z));
	} else {
		tmp = fma((z / (z - a)), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.9e+21)
		tmp = fma(Float64(1.0 - Float64(y / z)), t, x);
	elseif (z <= 5.1e-11)
		tmp = Float64(x + Float64(Float64(t * y) / Float64(a - z)));
	else
		tmp = fma(Float64(z / Float64(z - a)), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.9e+21], N[(N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 5.1e-11], N[(x + N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.9e21

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.7%

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

   6. Applied rewrites 67.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 67.7%

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

   8. Applied rewrites 67.7%

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

   if -3.9e21 < z < 5.0999999999999998e-11

   1. Initial program 85.4%

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

   2. Taylor expanded in y around inf

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

      1. lower-*.f64 73.2%

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

   4. Applied rewrites 73.2%

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

   if 5.0999999999999998e-11 < z

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 72.5%

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

Alternative 6: 82.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-199}:\\ \;\;\;\;\mathsf{fma}\left(\frac{1}{a} \cdot y, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, t, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= z -2.2e-64)
  (fma (- 1.0 (/ y z)) t x)
  (if (<= z 6.5e-199)
    (fma (* (/ 1.0 a) y) t x)
    (fma (/ z (- z a)) t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-64) {
		tmp = fma((1.0 - (y / z)), t, x);
	} else if (z <= 6.5e-199) {
		tmp = fma(((1.0 / a) * y), t, x);
	} else {
		tmp = fma((z / (z - a)), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e-64)
		tmp = fma(Float64(1.0 - Float64(y / z)), t, x);
	elseif (z <= 6.5e-199)
		tmp = fma(Float64(Float64(1.0 / a) * y), t, x);
	else
		tmp = fma(Float64(z / Float64(z - a)), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e-64], N[(N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 6.5e-199], N[(N[(N[(1.0 / a), $MachinePrecision] * y), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e-64

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.7%

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

   6. Applied rewrites 67.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 67.7%

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

   8. Applied rewrites 67.7%

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

   if -2.2e-64 < z < 6.5000000000000002e-199

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 62.2%

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

   6. Applied rewrites 62.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 62.2%

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

   8. Applied rewrites 62.2%

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

   if 6.5000000000000002e-199 < z

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 72.5%

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

Alternative 7: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(1 - \frac{y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-199}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, t, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= z -2.2e-64)
  (fma (- 1.0 (/ y z)) t x)
  (if (<= z 6.5e-199) (fma (/ y a) t x) (fma (/ z (- z a)) t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-64) {
		tmp = fma((1.0 - (y / z)), t, x);
	} else if (z <= 6.5e-199) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = fma((z / (z - a)), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e-64)
		tmp = fma(Float64(1.0 - Float64(y / z)), t, x);
	elseif (z <= 6.5e-199)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = fma(Float64(z / Float64(z - a)), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e-64], N[(N[(1.0 - N[(y / z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 6.5e-199], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e-64

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.7%

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

   6. Applied rewrites 67.7%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. div-sub N/A

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

      4. *-inverses N/A

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

      5. lower--.f64 N/A

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

      6. lower-/.f64 67.7%

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

   8. Applied rewrites 67.7%

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

   if -2.2e-64 < z < 6.5000000000000002e-199

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 62.2%

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

   6. Applied rewrites 62.2%

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

   if 6.5000000000000002e-199 < z

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 72.5%

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

Alternative 8: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-64}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{z}, t, x\right)\\ \mathbf{elif}\;z \leq 6.5 \cdot 10^{-199}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, t, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= z -2.2e-64)
  (fma (/ (- z y) z) t x)
  (if (<= z 6.5e-199) (fma (/ y a) t x) (fma (/ z (- z a)) t x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-64) {
		tmp = fma(((z - y) / z), t, x);
	} else if (z <= 6.5e-199) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = fma((z / (z - a)), t, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e-64)
		tmp = fma(Float64(Float64(z - y) / z), t, x);
	elseif (z <= 6.5e-199)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = fma(Float64(z / Float64(z - a)), t, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e-64], N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t + x), $MachinePrecision], If[LessEqual[z, 6.5e-199], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.2e-64

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.7%

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

   6. Applied rewrites 67.7%

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

   if -2.2e-64 < z < 6.5000000000000002e-199

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 62.2%

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

   6. Applied rewrites 62.2%

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

   if 6.5000000000000002e-199 < z

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 72.5%

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

Alternative 9: 80.7% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{z}, t, x\right)\\ \mathbf{if}\;z \leq -2.2 \cdot 10^{-64}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 8.8 \cdot 10^{-17}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma (/ (- z y) z) t x)))
  (if (<= z -2.2e-64) t_1 (if (<= z 8.8e-17) (fma (/ y a) t x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / z), t, x);
	double tmp;
	if (z <= -2.2e-64) {
		tmp = t_1;
	} else if (z <= 8.8e-17) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / z), t, x)
	tmp = 0.0
	if (z <= -2.2e-64)
		tmp = t_1;
	elseif (z <= 8.8e-17)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / z), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[z, -2.2e-64], t$95$1, If[LessEqual[z, 8.8e-17], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -2.2e-64 or 8.8e-17 < z

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.7%

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

   6. Applied rewrites 67.7%

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

   if -2.2e-64 < z < 8.8e-17

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 62.2%

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

   6. Applied rewrites 62.2%

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

Alternative 10: 78.1% accurate, 0.9× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq -4.2 \cdot 10^{+21}:\\ \;\;\;\;x + t\\ \mathbf{elif}\;z \leq 5.1 \cdot 10^{-11}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{a}, t, x\right)\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= z -4.2e+21)
  (+ x t)
  (if (<= z 5.1e-11) (fma (/ y a) t x) (+ x t))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4.2e+21) {
		tmp = x + t;
	} else if (z <= 5.1e-11) {
		tmp = fma((y / a), t, x);
	} else {
		tmp = x + t;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4.2e+21)
		tmp = Float64(x + t);
	elseif (z <= 5.1e-11)
		tmp = fma(Float64(y / a), t, x);
	else
		tmp = Float64(x + t);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4.2e+21], N[(x + t), $MachinePrecision], If[LessEqual[z, 5.1e-11], N[(N[(y / a), $MachinePrecision] * t + x), $MachinePrecision], N[(x + t), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -4.2e21 or 5.0999999999999998e-11 < z

   1. Initial program 85.4%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{t} \]
   3. Step-by-step derivation

   4. Applied rewrites 61.8%

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

   if -4.2e21 < z < 5.0999999999999998e-11

   1. Initial program 85.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. add-flip N/A

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

      4. sub-flip N/A

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

      5. lift-/.f64 N/A

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

      6. lift-*.f64 N/A

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

      7. associate-*l/ N/A

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

      8. mult-flip-rev N/A

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

      9. remove-double-neg N/A

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

      10. lower-fma.f64 N/A

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

      11. mult-flip-rev N/A

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

      12. frac-2neg N/A

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

      13. lift--.f64 N/A

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

      14. sub-negate-rev N/A

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

      15. lower-/.f64 N/A

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

      16. lower--.f64 N/A

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

      17. lift--.f64 N/A

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

      18. sub-negate-rev N/A

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

      19. lower--.f64 98.4%

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

   3. Applied rewrites 98.4%

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

   4. Taylor expanded in z around 0

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

      1. lower-/.f64 62.2%

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

   6. Applied rewrites 62.2%

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

Alternative 11: 61.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} \mathbf{if}\;y \leq -1.05 \cdot 10^{+243}:\\ \;\;\;\;\frac{t \cdot y}{a}\\ \mathbf{else}:\\ \;\;\;\;x + t\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= y -1.05e+243) (/ (* t y) a) (+ x t)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.05e+243) {
		tmp = (t * y) / a;
	} else {
		tmp = x + t;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (y <= (-1.05d+243)) then
        tmp = (t * y) / a
    else
        tmp = x + t
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (y <= -1.05e+243) {
		tmp = (t * y) / a;
	} else {
		tmp = x + t;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if y <= -1.05e+243:
		tmp = (t * y) / a
	else:
		tmp = x + t
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (y <= -1.05e+243)
		tmp = Float64(Float64(t * y) / a);
	else
		tmp = Float64(x + t);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (y <= -1.05e+243)
		tmp = (t * y) / a;
	else
		tmp = x + t;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[y, -1.05e+243], N[(N[(t * y), $MachinePrecision] / a), $MachinePrecision], N[(x + t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if y < -1.05e243

   1. Initial program 85.4%

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

   2. Taylor expanded in x around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 38.6%

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

   4. Applied rewrites 38.6%

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

   5. Taylor expanded in z around 0

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

   7. Applied rewrites 21.6%

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

   8. Taylor expanded in y around inf

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

      1. lower-*.f64 18.2%

         \[\leadsto \frac{t \cdot y}{a} \]

   10. Applied rewrites 18.2%

       \[\leadsto \frac{t \cdot y}{a} \]

   if -1.05e243 < y

   1. Initial program 85.4%

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

   2. Taylor expanded in z around inf

      \[\leadsto x + \color{blue}{t} \]
   3. Step-by-step derivation

   4. Applied rewrites 61.8%

      \[\leadsto x + \color{blue}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 61.8% accurate, 4.3× speedup

Math

\[x + t \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 85.4%

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

2. Taylor expanded in z around inf

   \[\leadsto x + \color{blue}{t} \]
3. Step-by-step derivation

4. Applied rewrites 61.8%

   \[\leadsto x + \color{blue}{t} \]
5. Add Preprocessing

Alternative 13: 19.1% accurate, 15.6× speedup

Math

\[t \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

    interface fmax
        module procedure fmax88
        module procedure fmax44
        module procedure fmax84
        module procedure fmax48
    end interface
    interface fmin
        module procedure fmin88
        module procedure fmin44
        module procedure fmin84
        module procedure fmin48
    end interface
contains
    real(8) function fmax88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(4) function fmax44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, max(x, y), y /= y), x /= x)
    end function
    real(8) function fmax84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, max(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmax48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), max(dble(x), y), y /= y), x /= x)
    end function
    real(8) function fmin88(x, y) result (res)
        real(8), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(4) function fmin44(x, y) result (res)
        real(4), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(y, merge(x, min(x, y), y /= y), x /= x)
    end function
    real(8) function fmin84(x, y) result(res)
        real(8), intent (in) :: x
        real(4), intent (in) :: y
        res = merge(dble(y), merge(x, min(x, dble(y)), y /= y), x /= x)
    end function
    real(8) function fmin48(x, y) result(res)
        real(4), intent (in) :: x
        real(8), intent (in) :: y
        res = merge(y, merge(dble(x), min(dble(x), y), y /= y), x /= x)
    end function
end module

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

code[x_, y_, z_, t_, a_] := t

Derivation

1. Initial program 85.4%

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

2. Taylor expanded in x around 0

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

   1. lower-/.f64 N/A

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

   2. lower-*.f64 N/A

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

   3. lower--.f64 N/A

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

   4. lower--.f64 38.6%

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

4. Applied rewrites 38.6%

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

5. Taylor expanded in z around 0

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

7. Applied rewrites 21.6%

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

8. Taylor expanded in z around inf

   \[\leadsto t \]
9. Step-by-step derivation

10. Applied rewrites 19.1%

    \[\leadsto t \]
11. Add Preprocessing

Reproduce

herbie shell --seed 2025212 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Plot.Render.Plot.Axis:renderAxisTick from plot-0.2.3.4, A"
  :precision binary64
  (+ x (/ (* (- y z) t) (- a z))))