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

Percentage Accurate: 98.1% → 98.1%

Time: 6.7s

Alternatives: 11

Speedup: 1.0×

Specification

Math

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

FPCore

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

C

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

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) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 98.1% accurate, 1.0× speedup

Math

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

FPCore

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

C

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

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) / (z - a)))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 98.1% 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(Float64(t - z) / Float64(a - z)), y, x)
end

Wolfram

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

Derivation

1. Initial program 98.1%

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

   1. lift-+.f64 N/A

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

   2. +-commutative N/A

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

   3. lift-*.f64 N/A

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

   4. *-commutative N/A

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

   5. lower-fma.f64 98.1%

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

   6. lift-/.f64 N/A

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

   7. frac-2neg N/A

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

   8. lower-/.f64 N/A

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

   9. lift--.f64 N/A

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

   10. sub-negate-rev N/A

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

   11. lower--.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-negate-rev N/A

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

   14. lower--.f64 98.1%

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

3. Applied rewrites 98.1%

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

Alternative 2: 96.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -20000000:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t - z}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 1.00000002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ t (- a z)) y x)))
  (if (<= t_1 -20000000.0)
    t_2
    (if (<= t_1 5e-47)
      (fma (/ (- t z) a) y x)
      (if (<= t_1 1.00000002) (fma (/ z (- z a)) y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((t / (a - z)), y, x);
	double tmp;
	if (t_1 <= -20000000.0) {
		tmp = t_2;
	} else if (t_1 <= 5e-47) {
		tmp = fma(((t - z) / a), y, x);
	} else if (t_1 <= 1.00000002) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(t / Float64(a - z)), y, x)
	tmp = 0.0
	if (t_1 <= -20000000.0)
		tmp = t_2;
	elseif (t_1 <= 5e-47)
		tmp = fma(Float64(Float64(t - z) / a), y, x);
	elseif (t_1 <= 1.00000002)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -20000000.0], t$95$2, If[LessEqual[t$95$1, 5e-47], N[(N[(N[(t - z), $MachinePrecision] / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1.00000002], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -2e7 or 1.0000000200000001 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.1%

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.3%

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

   if -2e7 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.0000000000000001e-47

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.1%

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 60.7%

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

   if 5.0000000000000001e-47 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000200000001

   1. Initial program 98.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.4%

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

   4. Applied rewrites 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4%

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

   6. Applied rewrites 67.4%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 72.0%

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

   9. Applied rewrites 72.0%

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

Alternative 3: 96.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;z \leq 1.15 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{z - a}, z - t, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (if (<= z 1.15e+134)
  (fma (/ y (- z a)) (- z t) x)
  (fma (/ (- z t) z) y x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= 1.15e+134) {
		tmp = fma((y / (z - a)), (z - t), x);
	} else {
		tmp = fma(((z - t) / z), y, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= 1.15e+134)
		tmp = fma(Float64(y / Float64(z - a)), Float64(z - t), x);
	else
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, 1.15e+134], N[(N[(y / N[(z - a), $MachinePrecision]), $MachinePrecision] * N[(z - t), $MachinePrecision] + x), $MachinePrecision], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < 1.1499999999999999e134

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lift-/.f64 N/A

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

      6. mult-flip N/A

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

      7. associate-*l* N/A

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

      8. *-commutative N/A

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

      9. lower-fma.f64 N/A

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

      10. *-commutative N/A

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

      11. mult-flip-rev N/A

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

      12. lower-/.f64 96.0%

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

   3. Applied rewrites 96.0%

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

   if 1.1499999999999999e134 < z

   1. Initial program 98.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.4%

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

   4. Applied rewrites 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4%

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

   6. Applied rewrites 67.4%

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

Alternative 4: 93.2% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{t}{a - z}, y, x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-72}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.00000002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ t (- a z)) y x)))
  (if (<= t_1 -1e-72)
    t_2
    (if (<= t_1 1.00000002) (fma (/ z (- z a)) y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((t / (a - z)), y, x);
	double tmp;
	if (t_1 <= -1e-72) {
		tmp = t_2;
	} else if (t_1 <= 1.00000002) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(t / Float64(a - z)), y, x)
	tmp = 0.0
	if (t_1 <= -1e-72)
		tmp = t_2;
	elseif (t_1 <= 1.00000002)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / N[(a - z), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-72], t$95$2, If[LessEqual[t$95$1, 1.00000002], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -9.9999999999999997e-73 or 1.0000000200000001 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.1%

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.3%

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

   if -9.9999999999999997e-73 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000200000001

   1. Initial program 98.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.4%

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

   4. Applied rewrites 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4%

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

   6. Applied rewrites 67.4%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 72.0%

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

   9. Applied rewrites 72.0%

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

Alternative 5: 92.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{y}{a - z}, t, x\right)\\ \mathbf{if}\;t\_1 \leq -1 \cdot 10^{-72}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 1.00000002:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ y (- a z)) t x)))
  (if (<= t_1 -1e-72)
    t_2
    (if (<= t_1 1.00000002) (fma (/ z (- z a)) y x) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((y / (a - z)), t, x);
	double tmp;
	if (t_1 <= -1e-72) {
		tmp = t_2;
	} else if (t_1 <= 1.00000002) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(y / Float64(a - z)), t, x)
	tmp = 0.0
	if (t_1 <= -1e-72)
		tmp = t_2;
	elseif (t_1 <= 1.00000002)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(y / N[(a - z), $MachinePrecision]), $MachinePrecision] * t + x), $MachinePrecision]}, If[LessEqual[t$95$1, -1e-72], t$95$2, If[LessEqual[t$95$1, 1.00000002], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -9.9999999999999997e-73 or 1.0000000200000001 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.1%

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.3%

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

      1. lift-fma.f64 N/A

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

      2. add-flip N/A

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

      3. sub-flip N/A

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

      4. lift-/.f64 N/A

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

      5. associate-*l/ N/A

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

      6. associate-/l* N/A

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

      7. *-commutative N/A

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

      8. remove-double-neg N/A

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

      9. lower-fma.f64 N/A

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

      10. lower-/.f64 76.5%

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

   8. Applied rewrites 76.5%

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

   if -9.9999999999999997e-73 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.0000000200000001

   1. Initial program 98.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.4%

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

   4. Applied rewrites 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4%

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

   6. Applied rewrites 67.4%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 72.0%

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

   9. Applied rewrites 72.0%

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

Alternative 6: 84.9% accurate, 0.2× speedup

Math

\[\begin{array}{l} t_1 := \mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ t_2 := \frac{z - t}{z - a}\\ t_3 := \frac{t \cdot y}{a - z}\\ \mathbf{if}\;t\_2 \leq -4 \cdot 10^{+109}:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq -1 \cdot 10^{-72}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 10^{-186}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{-16}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+134}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - t}{z}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (fma (/ t a) y x))
       (t_2 (/ (- z t) (- z a)))
       (t_3 (/ (* t y) (- a z))))
  (if (<= t_2 -4e+109)
    t_3
    (if (<= t_2 -1e-72)
      t_1
      (if (<= t_2 1e-186)
        (fma (/ z (- z a)) y x)
        (if (<= t_2 2e-16)
          t_1
          (if (<= t_2 2e+134) (fma (/ (- z t) z) y x) t_3)))))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((t / a), y, x);
	double t_2 = (z - t) / (z - a);
	double t_3 = (t * y) / (a - z);
	double tmp;
	if (t_2 <= -4e+109) {
		tmp = t_3;
	} else if (t_2 <= -1e-72) {
		tmp = t_1;
	} else if (t_2 <= 1e-186) {
		tmp = fma((z / (z - a)), y, x);
	} else if (t_2 <= 2e-16) {
		tmp = t_1;
	} else if (t_2 <= 2e+134) {
		tmp = fma(((z - t) / z), y, x);
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(t / a), y, x)
	t_2 = Float64(Float64(z - t) / Float64(z - a))
	t_3 = Float64(Float64(t * y) / Float64(a - z))
	tmp = 0.0
	if (t_2 <= -4e+109)
		tmp = t_3;
	elseif (t_2 <= -1e-72)
		tmp = t_1;
	elseif (t_2 <= 1e-186)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	elseif (t_2 <= 2e-16)
		tmp = t_1;
	elseif (t_2 <= 2e+134)
		tmp = fma(Float64(Float64(z - t) / z), y, x);
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]}, Block[{t$95$2 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -4e+109], t$95$3, If[LessEqual[t$95$2, -1e-72], t$95$1, If[LessEqual[t$95$2, 1e-186], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$2, 2e-16], t$95$1, If[LessEqual[t$95$2, 2e+134], N[(N[(N[(z - t), $MachinePrecision] / z), $MachinePrecision] * y + x), $MachinePrecision], t$95$3]]]]]]]]

Derivation

1. Split input into 4 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.9999999999999999e109 or 1.9999999999999998e134 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.1%

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.3%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 25.9%

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

   9. Applied rewrites 25.9%

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

   if -3.9999999999999999e109 < (/.f64 (-.f64 z t) (-.f64 z a)) < -9.9999999999999997e-73 or 9.9999999999999991e-187 < (/.f64 (-.f64 z t) (-.f64 z a)) < 2e-16

   1. Initial program 98.1%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 62.2%

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

   4. Applied rewrites 62.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 62.2%

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

   6. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 62.2%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 62.2%

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

   8. Applied rewrites 62.2%

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

   if -9.9999999999999997e-73 < (/.f64 (-.f64 z t) (-.f64 z a)) < 9.9999999999999991e-187

   1. Initial program 98.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.4%

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

   4. Applied rewrites 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4%

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

   6. Applied rewrites 67.4%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 72.0%

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

   9. Applied rewrites 72.0%

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

   if 2e-16 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1.9999999999999998e134

   1. Initial program 98.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.4%

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

   4. Applied rewrites 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4%

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

   6. Applied rewrites 67.4%

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

Alternative 7: 84.5% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{t \cdot y}{a - z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+109}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq -1 \cdot 10^{-72}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+27}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{z - a}, y, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* t y) (- a z))))
  (if (<= t_1 -4e+109)
    t_2
    (if (<= t_1 -1e-72)
      (fma (/ t a) y x)
      (if (<= t_1 1e+27) (fma (/ z (- z a)) y x) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (t * y) / (a - z);
	double tmp;
	if (t_1 <= -4e+109) {
		tmp = t_2;
	} else if (t_1 <= -1e-72) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 1e+27) {
		tmp = fma((z / (z - a)), y, x);
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(t * y) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -4e+109)
		tmp = t_2;
	elseif (t_1 <= -1e-72)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 1e+27)
		tmp = fma(Float64(z / Float64(z - a)), y, x);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+109], t$95$2, If[LessEqual[t$95$1, -1e-72], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+27], N[(N[(z / N[(z - a), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.9999999999999999e109 or 1e27 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.1%

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.3%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 25.9%

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

   9. Applied rewrites 25.9%

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

   if -3.9999999999999999e109 < (/.f64 (-.f64 z t) (-.f64 z a)) < -9.9999999999999997e-73

   1. Initial program 98.1%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 62.2%

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

   4. Applied rewrites 62.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 62.2%

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

   6. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 62.2%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 62.2%

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

   8. Applied rewrites 62.2%

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

   if -9.9999999999999997e-73 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e27

   1. Initial program 98.1%

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

   2. Taylor expanded in a around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 67.4%

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

   4. Applied rewrites 67.4%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 67.4%

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

   6. Applied rewrites 67.4%

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

   7. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 72.0%

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

   9. Applied rewrites 72.0%

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

Alternative 8: 83.4% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \frac{t \cdot y}{a - z}\\ \mathbf{if}\;t\_1 \leq -4 \cdot 10^{+109}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{-47}:\\ \;\;\;\;\mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{elif}\;t\_1 \leq 10^{+27}:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (/ (- z t) (- z a))) (t_2 (/ (* t y) (- a z))))
  (if (<= t_1 -4e+109)
    t_2
    (if (<= t_1 5e-47)
      (fma (/ t a) y x)
      (if (<= t_1 1e+27) (+ x y) t_2)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = (t * y) / (a - z);
	double tmp;
	if (t_1 <= -4e+109) {
		tmp = t_2;
	} else if (t_1 <= 5e-47) {
		tmp = fma((t / a), y, x);
	} else if (t_1 <= 1e+27) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = Float64(Float64(t * y) / Float64(a - z))
	tmp = 0.0
	if (t_1 <= -4e+109)
		tmp = t_2;
	elseif (t_1 <= 5e-47)
		tmp = fma(Float64(t / a), y, x);
	elseif (t_1 <= 1e+27)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t * y), $MachinePrecision] / N[(a - z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$1, -4e+109], t$95$2, If[LessEqual[t$95$1, 5e-47], N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision], If[LessEqual[t$95$1, 1e+27], N[(x + y), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < -3.9999999999999999e109 or 1e27 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.1%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 98.1%

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

      6. lift-/.f64 N/A

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

      7. frac-2neg N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 N/A

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

      10. sub-negate-rev N/A

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

      11. lower--.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 98.1%

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

   3. Applied rewrites 98.1%

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

   4. Taylor expanded in z around 0

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

   6. Applied rewrites 76.3%

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

   7. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 25.9%

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

   9. Applied rewrites 25.9%

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

   if -3.9999999999999999e109 < (/.f64 (-.f64 z t) (-.f64 z a)) < 5.0000000000000001e-47

   1. Initial program 98.1%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 62.2%

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

   4. Applied rewrites 62.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 62.2%

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

   6. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 62.2%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 62.2%

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

   8. Applied rewrites 62.2%

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

   if 5.0000000000000001e-47 < (/.f64 (-.f64 z t) (-.f64 z a)) < 1e27

   1. Initial program 98.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-+.f64 60.7%

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

   4. Applied rewrites 60.7%

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

Alternative 9: 80.8% accurate, 0.4× speedup

Math

\[\begin{array}{l} t_1 := \frac{z - t}{z - a}\\ t_2 := \mathsf{fma}\left(\frac{t}{a}, y, x\right)\\ \mathbf{if}\;t\_1 \leq 5 \cdot 10^{-47}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq 30000:\\ \;\;\;\;x + y\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t a)
  :precision binary64
  (let* ((t_1 (/ (- z t) (- z a))) (t_2 (fma (/ t a) y x)))
  (if (<= t_1 5e-47) t_2 (if (<= t_1 30000.0) (+ x y) t_2))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (z - t) / (z - a);
	double t_2 = fma((t / a), y, x);
	double tmp;
	if (t_1 <= 5e-47) {
		tmp = t_2;
	} else if (t_1 <= 30000.0) {
		tmp = x + y;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(z - t) / Float64(z - a))
	t_2 = fma(Float64(t / a), y, x)
	tmp = 0.0
	if (t_1 <= 5e-47)
		tmp = t_2;
	elseif (t_1 <= 30000.0)
		tmp = Float64(x + y);
	else
		tmp = t_2;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z - t), $MachinePrecision] / N[(z - a), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(t / a), $MachinePrecision] * y + x), $MachinePrecision]}, If[LessEqual[t$95$1, 5e-47], t$95$2, If[LessEqual[t$95$1, 30000.0], N[(x + y), $MachinePrecision], t$95$2]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 z t) (-.f64 z a)) < 5.0000000000000001e-47 or 3e4 < (/.f64 (-.f64 z t) (-.f64 z a))

   1. Initial program 98.1%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 62.2%

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

   4. Applied rewrites 62.2%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. *-commutative N/A

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

      4. lower-*.f64 N/A

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

      5. lower-/.f64 62.2%

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

   6. Applied rewrites 62.2%

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

      1. lift-+.f64 N/A

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

      2. +-commutative N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. lower-fma.f64 62.2%

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

      6. lift-*.f64 N/A

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

      7. *-commutative N/A

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

      8. lift-/.f64 N/A

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

      9. mult-flip-rev N/A

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

      10. lower-/.f64 62.2%

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

   8. Applied rewrites 62.2%

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

   if 5.0000000000000001e-47 < (/.f64 (-.f64 z t) (-.f64 z a)) < 3e4

   1. Initial program 98.1%

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

   2. Taylor expanded in z around inf

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

      1. lower-+.f64 60.7%

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

   4. Applied rewrites 60.7%

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

Alternative 10: 60.7% accurate, 4.3× speedup

Math

\[x + y \]

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in z around inf

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

   1. lower-+.f64 60.7%

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

4. Applied rewrites 60.7%

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

Alternative 11: 19.1% accurate, 15.6× speedup

Math

\[y \]

FPCore

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

C

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

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 = y
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 98.1%

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

2. Taylor expanded in z around inf

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

   1. lower-+.f64 60.7%

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

4. Applied rewrites 60.7%

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

5. Taylor expanded in x around 0

   \[\leadsto y \]
6. Step-by-step derivation

7. Applied rewrites 19.1%

   \[\leadsto y \]
8. Add Preprocessing

Reproduce

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