Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2

Percentage Accurate: 86.5% → 99.2%

Time: 7.5s

Alternatives: 13

Speedup: 1.2×

• 23.3% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x / y) + ((2.0d0 + ((z * 2.0d0) * (1.0d0 - t))) / (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

Python

def code(x, y, z, t):
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x / y) + Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Initial Program: 86.5% accurate, 1.0× speedup

Math

\[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))

C

double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * 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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = (x / y) + ((2.0d0 + ((z * 2.0d0) * (1.0d0 - t))) / (t * z))
end function

Java

public static double code(double x, double y, double z, double t) {
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
}

Python

def code(x, y, z, t):
	return (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z))

Julia

function code(x, y, z, t)
	return Float64(Float64(x / y) + Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z)))
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = (x / y) + ((2.0 + ((z * 2.0) * (1.0 - t))) / (t * z));
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]

Alternative 1: 99.2% accurate, 1.2× speedup

Math

\[\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t} + \frac{x}{y} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (+ (/ (fma (- 1.0 t) 2.0 (/ 2.0 z)) t) (/ x y)))

C

double code(double x, double y, double z, double t) {
	return (fma((1.0 - t), 2.0, (2.0 / z)) / t) + (x / y);
}

Julia

function code(x, y, z, t)
	return Float64(Float64(fma(Float64(1.0 - t), 2.0, Float64(2.0 / z)) / t) + Float64(x / y))
end

Wolfram

code[x_, y_, z_, t_] := N[(N[(N[(N[(1.0 - t), $MachinePrecision] * 2.0 + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision]

Derivation

1. Initial program 86.5%

   \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
2. Step-by-step derivation

   1. lift-+.f64 N/A

      \[\leadsto \color{blue}{\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]

   2. +-commutative N/A

      \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

   3. lower-+.f64 86.5

      \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

   4. lift-/.f64 N/A

      \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} + \frac{x}{y} \]

   5. lift-*.f64 N/A

      \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

   6. *-commutative N/A

      \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{z \cdot t}} + \frac{x}{y} \]

   7. associate-/r* N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

   8. lower-/.f64 N/A

      \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

   9. lift-+.f64 N/A

      \[\leadsto \frac{\frac{\color{blue}{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}}{z}}{t} + \frac{x}{y} \]

   10. +-commutative N/A

       \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right) + 2}}{z}}{t} + \frac{x}{y} \]

   11. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right)} + 2}{z}}{t} + \frac{x}{y} \]

   12. lift-*.f64 N/A

       \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(1 - t\right) + 2}{z}}{t} + \frac{x}{y} \]

   13. associate-*l* N/A

       \[\leadsto \frac{\frac{\color{blue}{z \cdot \left(2 \cdot \left(1 - t\right)\right)} + 2}{z}}{t} + \frac{x}{y} \]

   14. *-commutative N/A

       \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \left(1 - t\right)\right) \cdot z} + 2}{z}}{t} + \frac{x}{y} \]

   15. add-to-fraction-rev N/A

       \[\leadsto \frac{\color{blue}{2 \cdot \left(1 - t\right) + \frac{2}{z}}}{t} + \frac{x}{y} \]

   16. *-commutative N/A

       \[\leadsto \frac{\color{blue}{\left(1 - t\right) \cdot 2} + \frac{2}{z}}{t} + \frac{x}{y} \]

   17. lower-fma.f64 N/A

       \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}}{t} + \frac{x}{y} \]

   18. lower-/.f64 99.2

       \[\leadsto \frac{\mathsf{fma}\left(1 - t, 2, \color{blue}{\frac{2}{z}}\right)}{t} + \frac{x}{y} \]

3. Applied rewrites 99.2%

   \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t} + \frac{x}{y}} \]
4. Add Preprocessing

Alternative 2: 98.2% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{x}{y} + \frac{2 + 2 \cdot z}{t \cdot z}\\ \mathbf{if}\;\frac{x}{y} \leq -50000:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 5 \cdot 10^{-6}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, 1 - t, \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (/ (+ 2.0 (* 2.0 z)) (* t z)))))
   (if (<= (/ x y) -50000.0)
     t_1
     (if (<= (/ x y) 5e-6) (/ (fma 2.0 (- 1.0 t) (/ 2.0 z)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + ((2.0 + (2.0 * z)) / (t * z));
	double tmp;
	if ((x / y) <= -50000.0) {
		tmp = t_1;
	} else if ((x / y) <= 5e-6) {
		tmp = fma(2.0, (1.0 - t), (2.0 / z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(Float64(2.0 + Float64(2.0 * z)) / Float64(t * z)))
	tmp = 0.0
	if (Float64(x / y) <= -50000.0)
		tmp = t_1;
	elseif (Float64(x / y) <= 5e-6)
		tmp = Float64(fma(2.0, Float64(1.0 - t), Float64(2.0 / z)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(N[(2.0 + N[(2.0 * z), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -50000.0], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 5e-6], N[(N[(2.0 * N[(1.0 - t), $MachinePrecision] + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -5e4 or 5.00000000000000041e-6 < (/.f64 x y)

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in t around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2 + 2 \cdot z}}{t \cdot z} \]
   3. Step-by-step derivation

      1. lower-+.f64 N/A

         \[\leadsto \frac{x}{y} + \frac{2 + \color{blue}{2 \cdot z}}{t \cdot z} \]

      2. lower-*.f64 80.3

         \[\leadsto \frac{x}{y} + \frac{2 + 2 \cdot \color{blue}{z}}{t \cdot z} \]

   4. Applied rewrites 80.3%

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2 + 2 \cdot z}}{t \cdot z} \]

   if -5e4 < (/.f64 x y) < 5.00000000000000041e-6

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{1 - t}{t}}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{\color{blue}{t}}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      6. lower-*.f64 66.5

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + \color{blue}{2 \cdot \frac{1}{t \cdot z}} \]

      2. sum-to-mult N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \color{blue}{\left(2 \cdot \frac{1 - t}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      5. mult-flip-rev N/A

         \[\leadsto \left(1 + \frac{\frac{2}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(1 + \frac{\frac{2}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(1 + \frac{\frac{2}{z \cdot t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      8. associate-/l/ N/A

         \[\leadsto \left(1 + \frac{\frac{\frac{2}{z}}{t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(1 + \frac{\frac{\frac{2}{z}}{t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      10. sum-to-mult N/A

          \[\leadsto 2 \cdot \frac{1 - t}{t} + \color{blue}{\frac{\frac{2}{z}}{t}} \]

      11. lift-/.f64 N/A

          \[\leadsto 2 \cdot \frac{1 - t}{t} + \frac{\frac{2}{\color{blue}{z}}}{t} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2 \cdot \left(1 - t\right)}{t} + \frac{\color{blue}{\frac{2}{z}}}{t} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(1 - t\right) \cdot 2}{t} + \frac{\frac{\color{blue}{2}}{z}}{t} \]

      14. div-add N/A

          \[\leadsto \frac{\left(1 - t\right) \cdot 2 + \frac{2}{z}}{\color{blue}{t}} \]

   6. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2, 1 - t, \frac{2}{z}\right)}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 92.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{t}, \frac{y}{z}, x\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, 1 - t, \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -5e+41)
   (/ (fma (/ 2.0 t) (/ y z) x) y)
   (if (<= (/ x y) 2e+35)
     (/ (fma 2.0 (- 1.0 t) (/ 2.0 z)) t)
     (+ (/ x y) (/ 2.0 (* t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -5e+41) {
		tmp = fma((2.0 / t), (y / z), x) / y;
	} else if ((x / y) <= 2e+35) {
		tmp = fma(2.0, (1.0 - t), (2.0 / z)) / t;
	} else {
		tmp = (x / y) + (2.0 / (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -5e+41)
		tmp = Float64(fma(Float64(2.0 / t), Float64(y / z), x) / y);
	elseif (Float64(x / y) <= 2e+35)
		tmp = Float64(fma(2.0, Float64(1.0 - t), Float64(2.0 / z)) / t);
	else
		tmp = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -5e+41], N[(N[(N[(2.0 / t), $MachinePrecision] * N[(y / z), $MachinePrecision] + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2e+35], N[(N[(2.0 * N[(1.0 - t), $MachinePrecision] + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -5.00000000000000022e41

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 62.3%

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z} + \frac{x}{y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot z} + \color{blue}{\frac{x}{y}} \]

      4. add-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{t \cdot z} \cdot y + x}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{t \cdot z} \cdot y + x}{y}} \]

      6. lower-fma.f64 56.9

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{t \cdot z}, y, x\right)}}{y} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{\color{blue}{t \cdot z}}, y, x\right)}{y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{\color{blue}{z \cdot t}}, y, x\right)}{y} \]

      9. lower-*.f64 56.9

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{\color{blue}{z \cdot t}}, y, x\right)}{y} \]

   6. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{2}{z \cdot t}, y, x\right)}{y}} \]
   7. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{z \cdot t} \cdot y + x}}{y} \]

      2. lift-/.f64 N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{z \cdot t}} \cdot y + x}{y} \]

      3. associate-*l/ N/A

         \[\leadsto \frac{\color{blue}{\frac{2 \cdot y}{z \cdot t}} + x}{y} \]

      4. lift-*.f64 N/A

         \[\leadsto \frac{\frac{2 \cdot y}{\color{blue}{z \cdot t}} + x}{y} \]

      5. *-commutative N/A

         \[\leadsto \frac{\frac{2 \cdot y}{\color{blue}{t \cdot z}} + x}{y} \]

      6. times-frac N/A

         \[\leadsto \frac{\color{blue}{\frac{2}{t} \cdot \frac{y}{z}} + x}{y} \]

      7. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{t}, \frac{y}{z}, x\right)}}{y} \]

      8. lower-/.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\color{blue}{\frac{2}{t}}, \frac{y}{z}, x\right)}{y} \]

      9. lower-/.f64 54.9

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{t}, \color{blue}{\frac{y}{z}}, x\right)}{y} \]

   8. Applied rewrites 54.9%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{t}, \frac{y}{z}, x\right)}}{y} \]

   if -5.00000000000000022e41 < (/.f64 x y) < 1.9999999999999999e35

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{1 - t}{t}}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{\color{blue}{t}}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      6. lower-*.f64 66.5

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + \color{blue}{2 \cdot \frac{1}{t \cdot z}} \]

      2. sum-to-mult N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \color{blue}{\left(2 \cdot \frac{1 - t}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      5. mult-flip-rev N/A

         \[\leadsto \left(1 + \frac{\frac{2}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(1 + \frac{\frac{2}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(1 + \frac{\frac{2}{z \cdot t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      8. associate-/l/ N/A

         \[\leadsto \left(1 + \frac{\frac{\frac{2}{z}}{t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(1 + \frac{\frac{\frac{2}{z}}{t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      10. sum-to-mult N/A

          \[\leadsto 2 \cdot \frac{1 - t}{t} + \color{blue}{\frac{\frac{2}{z}}{t}} \]

      11. lift-/.f64 N/A

          \[\leadsto 2 \cdot \frac{1 - t}{t} + \frac{\frac{2}{\color{blue}{z}}}{t} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2 \cdot \left(1 - t\right)}{t} + \frac{\color{blue}{\frac{2}{z}}}{t} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(1 - t\right) \cdot 2}{t} + \frac{\frac{\color{blue}{2}}{z}}{t} \]

      14. div-add N/A

          \[\leadsto \frac{\left(1 - t\right) \cdot 2 + \frac{2}{z}}{\color{blue}{t}} \]

   6. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2, 1 - t, \frac{2}{z}\right)}{t}} \]

   if 1.9999999999999999e35 < (/.f64 x y)

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 62.3%

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

Alternative 4: 92.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+41}:\\ \;\;\;\;\frac{\mathsf{fma}\left(\frac{2}{z \cdot t}, y, x\right)}{y}\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, 1 - t, \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;\frac{x}{y} + \frac{2}{t \cdot z}\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= (/ x y) -5e+41)
   (/ (fma (/ 2.0 (* z t)) y x) y)
   (if (<= (/ x y) 2e+35)
     (/ (fma 2.0 (- 1.0 t) (/ 2.0 z)) t)
     (+ (/ x y) (/ 2.0 (* t z))))))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if ((x / y) <= -5e+41) {
		tmp = fma((2.0 / (z * t)), y, x) / y;
	} else if ((x / y) <= 2e+35) {
		tmp = fma(2.0, (1.0 - t), (2.0 / z)) / t;
	} else {
		tmp = (x / y) + (2.0 / (t * z));
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (Float64(x / y) <= -5e+41)
		tmp = Float64(fma(Float64(2.0 / Float64(z * t)), y, x) / y);
	elseif (Float64(x / y) <= 2e+35)
		tmp = Float64(fma(2.0, Float64(1.0 - t), Float64(2.0 / z)) / t);
	else
		tmp = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[N[(x / y), $MachinePrecision], -5e+41], N[(N[(N[(2.0 / N[(z * t), $MachinePrecision]), $MachinePrecision] * y + x), $MachinePrecision] / y), $MachinePrecision], If[LessEqual[N[(x / y), $MachinePrecision], 2e+35], N[(N[(2.0 * N[(1.0 - t), $MachinePrecision] + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 x y) < -5.00000000000000022e41

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 62.3%

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   5. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2}{t \cdot z} + \frac{x}{y}} \]

      3. lift-/.f64 N/A

         \[\leadsto \frac{2}{t \cdot z} + \color{blue}{\frac{x}{y}} \]

      4. add-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{t \cdot z} \cdot y + x}{y}} \]

      5. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2}{t \cdot z} \cdot y + x}{y}} \]

      6. lower-fma.f64 56.9

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(\frac{2}{t \cdot z}, y, x\right)}}{y} \]

      7. lift-*.f64 N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{\color{blue}{t \cdot z}}, y, x\right)}{y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{\color{blue}{z \cdot t}}, y, x\right)}{y} \]

      9. lower-*.f64 56.9

         \[\leadsto \frac{\mathsf{fma}\left(\frac{2}{\color{blue}{z \cdot t}}, y, x\right)}{y} \]

   6. Applied rewrites 56.9%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(\frac{2}{z \cdot t}, y, x\right)}{y}} \]

   if -5.00000000000000022e41 < (/.f64 x y) < 1.9999999999999999e35

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{1 - t}{t}}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{\color{blue}{t}}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      6. lower-*.f64 66.5

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + \color{blue}{2 \cdot \frac{1}{t \cdot z}} \]

      2. sum-to-mult N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \color{blue}{\left(2 \cdot \frac{1 - t}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      5. mult-flip-rev N/A

         \[\leadsto \left(1 + \frac{\frac{2}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(1 + \frac{\frac{2}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(1 + \frac{\frac{2}{z \cdot t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      8. associate-/l/ N/A

         \[\leadsto \left(1 + \frac{\frac{\frac{2}{z}}{t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(1 + \frac{\frac{\frac{2}{z}}{t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      10. sum-to-mult N/A

          \[\leadsto 2 \cdot \frac{1 - t}{t} + \color{blue}{\frac{\frac{2}{z}}{t}} \]

      11. lift-/.f64 N/A

          \[\leadsto 2 \cdot \frac{1 - t}{t} + \frac{\frac{2}{\color{blue}{z}}}{t} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2 \cdot \left(1 - t\right)}{t} + \frac{\color{blue}{\frac{2}{z}}}{t} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(1 - t\right) \cdot 2}{t} + \frac{\frac{\color{blue}{2}}{z}}{t} \]

      14. div-add N/A

          \[\leadsto \frac{\left(1 - t\right) \cdot 2 + \frac{2}{z}}{\color{blue}{t}} \]

   6. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2, 1 - t, \frac{2}{z}\right)}{t}} \]

   if 1.9999999999999999e35 < (/.f64 x y)

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 62.3%

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

Alternative 5: 92.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_1 := \frac{x}{y} + \frac{2}{t \cdot z}\\ \mathbf{if}\;\frac{x}{y} \leq -5 \cdot 10^{+41}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;\frac{x}{y} \leq 2 \cdot 10^{+35}:\\ \;\;\;\;\frac{\mathsf{fma}\left(2, 1 - t, \frac{2}{z}\right)}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (+ (/ x y) (/ 2.0 (* t z)))))
   (if (<= (/ x y) -5e+41)
     t_1
     (if (<= (/ x y) 2e+35) (/ (fma 2.0 (- 1.0 t) (/ 2.0 z)) t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) + (2.0 / (t * z));
	double tmp;
	if ((x / y) <= -5e+41) {
		tmp = t_1;
	} else if ((x / y) <= 2e+35) {
		tmp = fma(2.0, (1.0 - t), (2.0 / z)) / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) + Float64(2.0 / Float64(t * z)))
	tmp = 0.0
	if (Float64(x / y) <= -5e+41)
		tmp = t_1;
	elseif (Float64(x / y) <= 2e+35)
		tmp = Float64(fma(2.0, Float64(1.0 - t), Float64(2.0 / z)) / t);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] + N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[(x / y), $MachinePrecision], -5e+41], t$95$1, If[LessEqual[N[(x / y), $MachinePrecision], 2e+35], N[(N[(2.0 * N[(1.0 - t), $MachinePrecision] + N[(2.0 / z), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 x y) < -5.00000000000000022e41 or 1.9999999999999999e35 < (/.f64 x y)

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in z around 0

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]
   3. Step-by-step derivation

   4. Applied rewrites 62.3%

      \[\leadsto \frac{x}{y} + \frac{\color{blue}{2}}{t \cdot z} \]

   if -5.00000000000000022e41 < (/.f64 x y) < 1.9999999999999999e35

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in x around 0

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + 2 \cdot \frac{1}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{1 - t}{t}}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{\color{blue}{t}}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      4. lower-*.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      5. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

      6. lower-*.f64 66.5

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right) \]

   4. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1 - t}{t}, 2 \cdot \frac{1}{t \cdot z}\right)} \]
   5. Step-by-step derivation

      1. lift-fma.f64 N/A

         \[\leadsto 2 \cdot \frac{1 - t}{t} + \color{blue}{2 \cdot \frac{1}{t \cdot z}} \]

      2. sum-to-mult N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \color{blue}{\left(2 \cdot \frac{1 - t}{t}\right)} \]

      3. lift-*.f64 N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      4. lift-/.f64 N/A

         \[\leadsto \left(1 + \frac{2 \cdot \frac{1}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      5. mult-flip-rev N/A

         \[\leadsto \left(1 + \frac{\frac{2}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      6. lift-*.f64 N/A

         \[\leadsto \left(1 + \frac{\frac{2}{t \cdot z}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      7. *-commutative N/A

         \[\leadsto \left(1 + \frac{\frac{2}{z \cdot t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      8. associate-/l/ N/A

         \[\leadsto \left(1 + \frac{\frac{\frac{2}{z}}{t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      9. lift-/.f64 N/A

         \[\leadsto \left(1 + \frac{\frac{\frac{2}{z}}{t}}{2 \cdot \frac{1 - t}{t}}\right) \cdot \left(2 \cdot \frac{1 - t}{t}\right) \]

      10. sum-to-mult N/A

          \[\leadsto 2 \cdot \frac{1 - t}{t} + \color{blue}{\frac{\frac{2}{z}}{t}} \]

      11. lift-/.f64 N/A

          \[\leadsto 2 \cdot \frac{1 - t}{t} + \frac{\frac{2}{\color{blue}{z}}}{t} \]

      12. associate-*r/ N/A

          \[\leadsto \frac{2 \cdot \left(1 - t\right)}{t} + \frac{\color{blue}{\frac{2}{z}}}{t} \]

      13. *-commutative N/A

          \[\leadsto \frac{\left(1 - t\right) \cdot 2}{t} + \frac{\frac{\color{blue}{2}}{z}}{t} \]

      14. div-add N/A

          \[\leadsto \frac{\left(1 - t\right) \cdot 2 + \frac{2}{z}}{\color{blue}{t}} \]

   6. Applied rewrites 66.5%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(2, 1 - t, \frac{2}{z}\right)}{t}} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 6: 84.3% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\frac{2}{z} - -2}{t}\\ t_2 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_3 := \frac{x}{y} - 2\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq -1.99999999:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq 2 \cdot 10^{+142}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{1}{t}, \frac{x}{y}\right)\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- (/ 2.0 z) -2.0) t))
        (t_2 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z)))
        (t_3 (- (/ x y) 2.0)))
   (if (<= t_2 -5e+52)
     t_1
     (if (<= t_2 -1.99999999)
       t_3
       (if (<= t_2 2e+142)
         (fma 2.0 (/ 1.0 t) (/ x y))
         (if (<= t_2 INFINITY) t_1 t_3))))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((2.0 / z) - -2.0) / t;
	double t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_3 = (x / y) - 2.0;
	double tmp;
	if (t_2 <= -5e+52) {
		tmp = t_1;
	} else if (t_2 <= -1.99999999) {
		tmp = t_3;
	} else if (t_2 <= 2e+142) {
		tmp = fma(2.0, (1.0 / t), (x / y));
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(2.0 / z) - -2.0) / t)
	t_2 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	t_3 = Float64(Float64(x / y) - 2.0)
	tmp = 0.0
	if (t_2 <= -5e+52)
		tmp = t_1;
	elseif (t_2 <= -1.99999999)
		tmp = t_3;
	elseif (t_2 <= 2e+142)
		tmp = fma(2.0, Float64(1.0 / t), Float64(x / y));
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x / y), $MachinePrecision] - 2.0), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+52], t$95$1, If[LessEqual[t$95$2, -1.99999999], t$95$3, If[LessEqual[t$95$2, 2e+142], N[(2.0 * N[(1.0 / t), $MachinePrecision] + N[(x / y), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5e52 or 2.0000000000000001e142 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      3. lower-+.f64 86.5

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} + \frac{x}{y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{z \cdot t}} + \frac{x}{y} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}}{z}}{t} + \frac{x}{y} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right) + 2}}{z}}{t} + \frac{x}{y} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right)} + 2}{z}}{t} + \frac{x}{y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(1 - t\right) + 2}{z}}{t} + \frac{x}{y} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\color{blue}{z \cdot \left(2 \cdot \left(1 - t\right)\right)} + 2}{z}}{t} + \frac{x}{y} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \left(1 - t\right)\right) \cdot z} + 2}{z}}{t} + \frac{x}{y} \]

      15. add-to-fraction-rev N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left(1 - t\right) + \frac{2}{z}}}{t} + \frac{x}{y} \]

      16. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 - t\right) \cdot 2} + \frac{2}{z}}{t} + \frac{x}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}}{t} + \frac{x}{y} \]

      18. lower-/.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, 2, \color{blue}{\frac{2}{z}}\right)}{t} + \frac{x}{y} \]

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t} + \frac{x}{y}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}} + \frac{x}{y} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right) \cdot \frac{1}{t}} + \frac{x}{y} \]

      3. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - t\right) \cdot 2 + \frac{2}{z}\right)} \cdot \frac{1}{t} + \frac{x}{y} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(1 - t\right) \cdot 2 + \color{blue}{\frac{2}{z}}\right) \cdot \frac{1}{t} + \frac{x}{y} \]

      5. add-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2}{z}} \cdot \frac{1}{t} + \frac{x}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2\right) \cdot \frac{1}{t}}{z}} + \frac{x}{y} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 - t\right) \cdot \left(2 \cdot z\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(1 - t\right) \cdot \color{blue}{\left(z \cdot 2\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, z \cdot 2, 2\right)} \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{2 \cdot z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      11. count-2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      13. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}}{z} + \frac{x}{y} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}{z}} + \frac{x}{y} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(z + z, 1 - t, 2\right)}{t}}{z}} + \frac{x}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{\color{blue}{t}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      4. lower-/.f64 48.3

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} + 2}{t} \]

      3. add-flip N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{\frac{2}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      9. metadata-eval 48.3

         \[\leadsto \frac{\frac{2}{z} - -2}{t} \]

   10. Applied rewrites 48.3%

       \[\leadsto \frac{\frac{2}{z} - -2}{\color{blue}{t}} \]

   if -5e52 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -1.9999999900000001 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x}{y} - \color{blue}{2} \]

      2. lower-/.f64 53.5

         \[\leadsto \frac{x}{y} - 2 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]

   if -1.9999999900000001 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 2.0000000000000001e142

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      3. lower-+.f64 86.5

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} + \frac{x}{y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{z \cdot t}} + \frac{x}{y} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}}{z}}{t} + \frac{x}{y} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right) + 2}}{z}}{t} + \frac{x}{y} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right)} + 2}{z}}{t} + \frac{x}{y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(1 - t\right) + 2}{z}}{t} + \frac{x}{y} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\color{blue}{z \cdot \left(2 \cdot \left(1 - t\right)\right)} + 2}{z}}{t} + \frac{x}{y} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \left(1 - t\right)\right) \cdot z} + 2}{z}}{t} + \frac{x}{y} \]

      15. add-to-fraction-rev N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left(1 - t\right) + \frac{2}{z}}}{t} + \frac{x}{y} \]

      16. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 - t\right) \cdot 2} + \frac{2}{z}}{t} + \frac{x}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}}{t} + \frac{x}{y} \]

      18. lower-/.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, 2, \color{blue}{\frac{2}{z}}\right)}{t} + \frac{x}{y} \]

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t} + \frac{x}{y}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}} + \frac{x}{y} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right) \cdot \frac{1}{t}} + \frac{x}{y} \]

      3. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - t\right) \cdot 2 + \frac{2}{z}\right)} \cdot \frac{1}{t} + \frac{x}{y} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(1 - t\right) \cdot 2 + \color{blue}{\frac{2}{z}}\right) \cdot \frac{1}{t} + \frac{x}{y} \]

      5. add-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2}{z}} \cdot \frac{1}{t} + \frac{x}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2\right) \cdot \frac{1}{t}}{z}} + \frac{x}{y} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 - t\right) \cdot \left(2 \cdot z\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(1 - t\right) \cdot \color{blue}{\left(z \cdot 2\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, z \cdot 2, 2\right)} \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{2 \cdot z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      11. count-2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      13. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}}{z} + \frac{x}{y} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}{z}} + \frac{x}{y} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(z + z, 1 - t, 2\right)}{t}}{z}} + \frac{x}{y} \]

   6. Taylor expanded in z around inf

      \[\leadsto \color{blue}{2 \cdot \frac{1 - t}{t} + \frac{x}{y}} \]
   7. Step-by-step derivation

      1. lower-fma.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{1 - t}{t}}, \frac{x}{y}\right) \]

      2. lower-/.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{\color{blue}{t}}, \frac{x}{y}\right) \]

      3. lower--.f64 N/A

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, \frac{x}{y}\right) \]

      4. lower-/.f64 71.5

         \[\leadsto \mathsf{fma}\left(2, \frac{1 - t}{t}, \frac{x}{y}\right) \]

   8. Applied rewrites 71.5%

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{1 - t}{t}, \frac{x}{y}\right)} \]

   9. Taylor expanded in t around 0

      \[\leadsto \mathsf{fma}\left(2, \frac{1}{t}, \frac{x}{y}\right) \]
   10. Step-by-step derivation

   11. Applied rewrites 53.1%

       \[\leadsto \mathsf{fma}\left(2, \frac{1}{t}, \frac{x}{y}\right) \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 7: 83.2% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{\frac{2}{z} - -2}{t}\\ t_2 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_3 := \frac{x}{y} - 2\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq 50000:\\ \;\;\;\;t\_3\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_3\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (- (/ 2.0 z) -2.0) t))
        (t_2 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z)))
        (t_3 (- (/ x y) 2.0)))
   (if (<= t_2 -5e+52)
     t_1
     (if (<= t_2 50000.0) t_3 (if (<= t_2 INFINITY) t_1 t_3)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = ((2.0 / z) - -2.0) / t;
	double t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_3 = (x / y) - 2.0;
	double tmp;
	if (t_2 <= -5e+52) {
		tmp = t_1;
	} else if (t_2 <= 50000.0) {
		tmp = t_3;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = ((2.0 / z) - -2.0) / t;
	double t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_3 = (x / y) - 2.0;
	double tmp;
	if (t_2 <= -5e+52) {
		tmp = t_1;
	} else if (t_2 <= 50000.0) {
		tmp = t_3;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_3;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = ((2.0 / z) - -2.0) / t
	t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	t_3 = (x / y) - 2.0
	tmp = 0
	if t_2 <= -5e+52:
		tmp = t_1
	elif t_2 <= 50000.0:
		tmp = t_3
	elif t_2 <= math.inf:
		tmp = t_1
	else:
		tmp = t_3
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(Float64(2.0 / z) - -2.0) / t)
	t_2 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	t_3 = Float64(Float64(x / y) - 2.0)
	tmp = 0.0
	if (t_2 <= -5e+52)
		tmp = t_1;
	elseif (t_2 <= 50000.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = ((2.0 / z) - -2.0) / t;
	t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	t_3 = (x / y) - 2.0;
	tmp = 0.0;
	if (t_2 <= -5e+52)
		tmp = t_1;
	elseif (t_2 <= 50000.0)
		tmp = t_3;
	elseif (t_2 <= Inf)
		tmp = t_1;
	else
		tmp = t_3;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(N[(2.0 / z), $MachinePrecision] - -2.0), $MachinePrecision] / t), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(N[(x / y), $MachinePrecision] - 2.0), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+52], t$95$1, If[LessEqual[t$95$2, 50000.0], t$95$3, If[LessEqual[t$95$2, Infinity], t$95$1, t$95$3]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5e52 or 5e4 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      3. lower-+.f64 86.5

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} + \frac{x}{y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{z \cdot t}} + \frac{x}{y} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}}{z}}{t} + \frac{x}{y} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right) + 2}}{z}}{t} + \frac{x}{y} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right)} + 2}{z}}{t} + \frac{x}{y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(1 - t\right) + 2}{z}}{t} + \frac{x}{y} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\color{blue}{z \cdot \left(2 \cdot \left(1 - t\right)\right)} + 2}{z}}{t} + \frac{x}{y} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \left(1 - t\right)\right) \cdot z} + 2}{z}}{t} + \frac{x}{y} \]

      15. add-to-fraction-rev N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left(1 - t\right) + \frac{2}{z}}}{t} + \frac{x}{y} \]

      16. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 - t\right) \cdot 2} + \frac{2}{z}}{t} + \frac{x}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}}{t} + \frac{x}{y} \]

      18. lower-/.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, 2, \color{blue}{\frac{2}{z}}\right)}{t} + \frac{x}{y} \]

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t} + \frac{x}{y}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}} + \frac{x}{y} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right) \cdot \frac{1}{t}} + \frac{x}{y} \]

      3. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - t\right) \cdot 2 + \frac{2}{z}\right)} \cdot \frac{1}{t} + \frac{x}{y} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(1 - t\right) \cdot 2 + \color{blue}{\frac{2}{z}}\right) \cdot \frac{1}{t} + \frac{x}{y} \]

      5. add-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2}{z}} \cdot \frac{1}{t} + \frac{x}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2\right) \cdot \frac{1}{t}}{z}} + \frac{x}{y} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 - t\right) \cdot \left(2 \cdot z\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(1 - t\right) \cdot \color{blue}{\left(z \cdot 2\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, z \cdot 2, 2\right)} \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{2 \cdot z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      11. count-2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      13. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}}{z} + \frac{x}{y} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}{z}} + \frac{x}{y} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(z + z, 1 - t, 2\right)}{t}}{z}} + \frac{x}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{\color{blue}{t}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      4. lower-/.f64 48.3

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   9. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      2. +-commutative N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} + 2}{t} \]

      3. add-flip N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      4. lower--.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      6. lift-/.f64 N/A

         \[\leadsto \frac{2 \cdot \frac{1}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      7. mult-flip-rev N/A

         \[\leadsto \frac{\frac{2}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      8. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{z} - \left(\mathsf{neg}\left(2\right)\right)}{t} \]

      9. metadata-eval 48.3

         \[\leadsto \frac{\frac{2}{z} - -2}{t} \]

   10. Applied rewrites 48.3%

       \[\leadsto \frac{\frac{2}{z} - -2}{\color{blue}{t}} \]

   if -5e52 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 5e4 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x}{y} - \color{blue}{2} \]

      2. lower-/.f64 53.5

         \[\leadsto \frac{x}{y} - 2 \]

   4. Applied rewrites 53.5%

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

Alternative 8: 67.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ t_2 := \frac{x}{y} - 2\\ \mathbf{if}\;t\_1 \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\frac{\frac{2}{z}}{t}\\ \mathbf{elif}\;t\_1 \leq 5 \cdot 10^{+155}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_1 \leq \infty:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z)))
        (t_2 (- (/ x y) 2.0)))
   (if (<= t_1 -5e+52)
     (/ (/ 2.0 z) t)
     (if (<= t_1 5e+155) t_2 (if (<= t_1 INFINITY) (/ (/ 2.0 t) z) t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_2 = (x / y) - 2.0;
	double tmp;
	if (t_1 <= -5e+52) {
		tmp = (2.0 / z) / t;
	} else if (t_1 <= 5e+155) {
		tmp = t_2;
	} else if (t_1 <= ((double) INFINITY)) {
		tmp = (2.0 / t) / z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double t_2 = (x / y) - 2.0;
	double tmp;
	if (t_1 <= -5e+52) {
		tmp = (2.0 / z) / t;
	} else if (t_1 <= 5e+155) {
		tmp = t_2;
	} else if (t_1 <= Double.POSITIVE_INFINITY) {
		tmp = (2.0 / t) / z;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	t_2 = (x / y) - 2.0
	tmp = 0
	if t_1 <= -5e+52:
		tmp = (2.0 / z) / t
	elif t_1 <= 5e+155:
		tmp = t_2
	elif t_1 <= math.inf:
		tmp = (2.0 / t) / z
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	t_2 = Float64(Float64(x / y) - 2.0)
	tmp = 0.0
	if (t_1 <= -5e+52)
		tmp = Float64(Float64(2.0 / z) / t);
	elseif (t_1 <= 5e+155)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = Float64(Float64(2.0 / t) / z);
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	t_2 = (x / y) - 2.0;
	tmp = 0.0;
	if (t_1 <= -5e+52)
		tmp = (2.0 / z) / t;
	elseif (t_1 <= 5e+155)
		tmp = t_2;
	elseif (t_1 <= Inf)
		tmp = (2.0 / t) / z;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] - 2.0), $MachinePrecision]}, If[LessEqual[t$95$1, -5e+52], N[(N[(2.0 / z), $MachinePrecision] / t), $MachinePrecision], If[LessEqual[t$95$1, 5e+155], t$95$2, If[LessEqual[t$95$1, Infinity], N[(N[(2.0 / t), $MachinePrecision] / z), $MachinePrecision], t$95$2]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5e52

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      3. lower-+.f64 86.5

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} + \frac{x}{y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{z \cdot t}} + \frac{x}{y} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}}{z}}{t} + \frac{x}{y} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right) + 2}}{z}}{t} + \frac{x}{y} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right)} + 2}{z}}{t} + \frac{x}{y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(1 - t\right) + 2}{z}}{t} + \frac{x}{y} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\color{blue}{z \cdot \left(2 \cdot \left(1 - t\right)\right)} + 2}{z}}{t} + \frac{x}{y} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \left(1 - t\right)\right) \cdot z} + 2}{z}}{t} + \frac{x}{y} \]

      15. add-to-fraction-rev N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left(1 - t\right) + \frac{2}{z}}}{t} + \frac{x}{y} \]

      16. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 - t\right) \cdot 2} + \frac{2}{z}}{t} + \frac{x}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}}{t} + \frac{x}{y} \]

      18. lower-/.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, 2, \color{blue}{\frac{2}{z}}\right)}{t} + \frac{x}{y} \]

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t} + \frac{x}{y}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}} + \frac{x}{y} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right) \cdot \frac{1}{t}} + \frac{x}{y} \]

      3. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - t\right) \cdot 2 + \frac{2}{z}\right)} \cdot \frac{1}{t} + \frac{x}{y} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(1 - t\right) \cdot 2 + \color{blue}{\frac{2}{z}}\right) \cdot \frac{1}{t} + \frac{x}{y} \]

      5. add-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2}{z}} \cdot \frac{1}{t} + \frac{x}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2\right) \cdot \frac{1}{t}}{z}} + \frac{x}{y} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 - t\right) \cdot \left(2 \cdot z\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(1 - t\right) \cdot \color{blue}{\left(z \cdot 2\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, z \cdot 2, 2\right)} \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{2 \cdot z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      11. count-2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      13. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}}{z} + \frac{x}{y} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}{z}} + \frac{x}{y} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(z + z, 1 - t, 2\right)}{t}}{z}} + \frac{x}{y} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lower-*.f64 30.4

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

   8. Applied rewrites 30.4%

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

      3. *-commutative N/A

         \[\leadsto \frac{2}{z \cdot \color{blue}{t}} \]

      4. associate-/l/ N/A

         \[\leadsto \frac{\frac{2}{z}}{\color{blue}{t}} \]

      5. lift-/.f64 N/A

         \[\leadsto \frac{\frac{2}{z}}{t} \]

      6. lift-/.f64 30.4

         \[\leadsto \frac{\frac{2}{z}}{\color{blue}{t}} \]

   10. Applied rewrites 30.4%

       \[\leadsto \frac{\frac{2}{z}}{\color{blue}{t}} \]

   if -5e52 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 4.9999999999999999e155 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x}{y} - \color{blue}{2} \]

      2. lower-/.f64 53.5

         \[\leadsto \frac{x}{y} - 2 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]

   if 4.9999999999999999e155 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      3. lower-+.f64 86.5

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} + \frac{x}{y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{z \cdot t}} + \frac{x}{y} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}}{z}}{t} + \frac{x}{y} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right) + 2}}{z}}{t} + \frac{x}{y} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right)} + 2}{z}}{t} + \frac{x}{y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(1 - t\right) + 2}{z}}{t} + \frac{x}{y} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\color{blue}{z \cdot \left(2 \cdot \left(1 - t\right)\right)} + 2}{z}}{t} + \frac{x}{y} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \left(1 - t\right)\right) \cdot z} + 2}{z}}{t} + \frac{x}{y} \]

      15. add-to-fraction-rev N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left(1 - t\right) + \frac{2}{z}}}{t} + \frac{x}{y} \]

      16. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 - t\right) \cdot 2} + \frac{2}{z}}{t} + \frac{x}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}}{t} + \frac{x}{y} \]

      18. lower-/.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, 2, \color{blue}{\frac{2}{z}}\right)}{t} + \frac{x}{y} \]

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t} + \frac{x}{y}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}} + \frac{x}{y} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right) \cdot \frac{1}{t}} + \frac{x}{y} \]

      3. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - t\right) \cdot 2 + \frac{2}{z}\right)} \cdot \frac{1}{t} + \frac{x}{y} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(1 - t\right) \cdot 2 + \color{blue}{\frac{2}{z}}\right) \cdot \frac{1}{t} + \frac{x}{y} \]

      5. add-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2}{z}} \cdot \frac{1}{t} + \frac{x}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2\right) \cdot \frac{1}{t}}{z}} + \frac{x}{y} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 - t\right) \cdot \left(2 \cdot z\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(1 - t\right) \cdot \color{blue}{\left(z \cdot 2\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, z \cdot 2, 2\right)} \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{2 \cdot z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      11. count-2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      13. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}}{z} + \frac{x}{y} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}{z}} + \frac{x}{y} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(z + z, 1 - t, 2\right)}{t}}{z}} + \frac{x}{y} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lower-*.f64 30.4

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

   8. Applied rewrites 30.4%

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{t}}{\color{blue}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2}{t}}{\color{blue}{z}} \]

      5. lower-/.f64 30.4

         \[\leadsto \frac{\frac{2}{t}}{z} \]

   10. Applied rewrites 30.4%

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

Alternative 9: 67.8% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ t_2 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_2 \leq -5 \cdot 10^{+52}:\\ \;\;\;\;\frac{2}{t \cdot z}\\ \mathbf{elif}\;t\_2 \leq 5 \cdot 10^{+155}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_2 \leq \infty:\\ \;\;\;\;\frac{\frac{2}{t}}{z}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ x y) 2.0))
        (t_2 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
   (if (<= t_2 -5e+52)
     (/ 2.0 (* t z))
     (if (<= t_2 5e+155) t_1 (if (<= t_2 INFINITY) (/ (/ 2.0 t) z) t_1)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) - 2.0;
	double t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_2 <= -5e+52) {
		tmp = 2.0 / (t * z);
	} else if (t_2 <= 5e+155) {
		tmp = t_1;
	} else if (t_2 <= ((double) INFINITY)) {
		tmp = (2.0 / t) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / y) - 2.0;
	double t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_2 <= -5e+52) {
		tmp = 2.0 / (t * z);
	} else if (t_2 <= 5e+155) {
		tmp = t_1;
	} else if (t_2 <= Double.POSITIVE_INFINITY) {
		tmp = (2.0 / t) / z;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / y) - 2.0
	t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	tmp = 0
	if t_2 <= -5e+52:
		tmp = 2.0 / (t * z)
	elif t_2 <= 5e+155:
		tmp = t_1
	elif t_2 <= math.inf:
		tmp = (2.0 / t) / z
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) - 2.0)
	t_2 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if (t_2 <= -5e+52)
		tmp = Float64(2.0 / Float64(t * z));
	elseif (t_2 <= 5e+155)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = Float64(Float64(2.0 / t) / z);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / y) - 2.0;
	t_2 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	tmp = 0.0;
	if (t_2 <= -5e+52)
		tmp = 2.0 / (t * z);
	elseif (t_2 <= 5e+155)
		tmp = t_1;
	elseif (t_2 <= Inf)
		tmp = (2.0 / t) / z;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] - 2.0), $MachinePrecision]}, Block[{t$95$2 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$2, -5e+52], N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$2, 5e+155], t$95$1, If[LessEqual[t$95$2, Infinity], N[(N[(2.0 / t), $MachinePrecision] / z), $MachinePrecision], t$95$1]]]]]

Derivation

1. Split input into 3 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5e52

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lower-*.f64 30.4

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

   4. Applied rewrites 30.4%

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]

   if -5e52 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 4.9999999999999999e155 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x}{y} - \color{blue}{2} \]

      2. lower-/.f64 53.5

         \[\leadsto \frac{x}{y} - 2 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]

   if 4.9999999999999999e155 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      3. lower-+.f64 86.5

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} + \frac{x}{y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{z \cdot t}} + \frac{x}{y} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}}{z}}{t} + \frac{x}{y} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right) + 2}}{z}}{t} + \frac{x}{y} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right)} + 2}{z}}{t} + \frac{x}{y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(1 - t\right) + 2}{z}}{t} + \frac{x}{y} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\color{blue}{z \cdot \left(2 \cdot \left(1 - t\right)\right)} + 2}{z}}{t} + \frac{x}{y} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \left(1 - t\right)\right) \cdot z} + 2}{z}}{t} + \frac{x}{y} \]

      15. add-to-fraction-rev N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left(1 - t\right) + \frac{2}{z}}}{t} + \frac{x}{y} \]

      16. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 - t\right) \cdot 2} + \frac{2}{z}}{t} + \frac{x}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}}{t} + \frac{x}{y} \]

      18. lower-/.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, 2, \color{blue}{\frac{2}{z}}\right)}{t} + \frac{x}{y} \]

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t} + \frac{x}{y}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}} + \frac{x}{y} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right) \cdot \frac{1}{t}} + \frac{x}{y} \]

      3. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - t\right) \cdot 2 + \frac{2}{z}\right)} \cdot \frac{1}{t} + \frac{x}{y} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(1 - t\right) \cdot 2 + \color{blue}{\frac{2}{z}}\right) \cdot \frac{1}{t} + \frac{x}{y} \]

      5. add-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2}{z}} \cdot \frac{1}{t} + \frac{x}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2\right) \cdot \frac{1}{t}}{z}} + \frac{x}{y} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 - t\right) \cdot \left(2 \cdot z\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(1 - t\right) \cdot \color{blue}{\left(z \cdot 2\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, z \cdot 2, 2\right)} \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{2 \cdot z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      11. count-2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      13. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}}{z} + \frac{x}{y} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}{z}} + \frac{x}{y} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(z + z, 1 - t, 2\right)}{t}}{z}} + \frac{x}{y} \]

   6. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lower-*.f64 30.4

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

   8. Applied rewrites 30.4%

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   9. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lift-*.f64 N/A

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

      3. associate-/r* N/A

         \[\leadsto \frac{\frac{2}{t}}{\color{blue}{z}} \]

      4. lower-/.f64 N/A

         \[\leadsto \frac{\frac{2}{t}}{\color{blue}{z}} \]

      5. lower-/.f64 30.4

         \[\leadsto \frac{\frac{2}{t}}{z} \]

   10. Applied rewrites 30.4%

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

Alternative 10: 67.7% accurate, 0.3× speedup

Math

\[\begin{array}{l} t_1 := \frac{2}{t \cdot z}\\ t_2 := \frac{x}{y} - 2\\ t_3 := \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}\\ \mathbf{if}\;t\_3 \leq -5 \cdot 10^{+52}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t\_3 \leq 2 \cdot 10^{+145}:\\ \;\;\;\;t\_2\\ \mathbf{elif}\;t\_3 \leq \infty:\\ \;\;\;\;t\_1\\ \mathbf{else}:\\ \;\;\;\;t\_2\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (/ 2.0 (* t z)))
        (t_2 (- (/ x y) 2.0))
        (t_3 (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))
   (if (<= t_3 -5e+52)
     t_1
     (if (<= t_3 2e+145) t_2 (if (<= t_3 INFINITY) t_1 t_2)))))

C

double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double t_2 = (x / y) - 2.0;
	double t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_3 <= -5e+52) {
		tmp = t_1;
	} else if (t_3 <= 2e+145) {
		tmp = t_2;
	} else if (t_3 <= ((double) INFINITY)) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = 2.0 / (t * z);
	double t_2 = (x / y) - 2.0;
	double t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	double tmp;
	if (t_3 <= -5e+52) {
		tmp = t_1;
	} else if (t_3 <= 2e+145) {
		tmp = t_2;
	} else if (t_3 <= Double.POSITIVE_INFINITY) {
		tmp = t_1;
	} else {
		tmp = t_2;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = 2.0 / (t * z)
	t_2 = (x / y) - 2.0
	t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z)
	tmp = 0
	if t_3 <= -5e+52:
		tmp = t_1
	elif t_3 <= 2e+145:
		tmp = t_2
	elif t_3 <= math.inf:
		tmp = t_1
	else:
		tmp = t_2
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(2.0 / Float64(t * z))
	t_2 = Float64(Float64(x / y) - 2.0)
	t_3 = Float64(Float64(2.0 + Float64(Float64(z * 2.0) * Float64(1.0 - t))) / Float64(t * z))
	tmp = 0.0
	if (t_3 <= -5e+52)
		tmp = t_1;
	elseif (t_3 <= 2e+145)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = 2.0 / (t * z);
	t_2 = (x / y) - 2.0;
	t_3 = (2.0 + ((z * 2.0) * (1.0 - t))) / (t * z);
	tmp = 0.0;
	if (t_3 <= -5e+52)
		tmp = t_1;
	elseif (t_3 <= 2e+145)
		tmp = t_2;
	elseif (t_3 <= Inf)
		tmp = t_1;
	else
		tmp = t_2;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(2.0 / N[(t * z), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(x / y), $MachinePrecision] - 2.0), $MachinePrecision]}, Block[{t$95$3 = N[(N[(2.0 + N[(N[(z * 2.0), $MachinePrecision] * N[(1.0 - t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t * z), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$3, -5e+52], t$95$1, If[LessEqual[t$95$3, 2e+145], t$95$2, If[LessEqual[t$95$3, Infinity], t$95$1, t$95$2]]]]]]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < -5e52 or 2e145 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < +inf.0

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in z around 0

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]
   3. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2}{\color{blue}{t \cdot z}} \]

      2. lower-*.f64 30.4

         \[\leadsto \frac{2}{t \cdot \color{blue}{z}} \]

   4. Applied rewrites 30.4%

      \[\leadsto \color{blue}{\frac{2}{t \cdot z}} \]

   if -5e52 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z)) < 2e145 or +inf.0 < (/.f64 (+.f64 #s(literal 2 binary64) (*.f64 (*.f64 z #s(literal 2 binary64)) (-.f64 #s(literal 1 binary64) t))) (*.f64 t z))

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x}{y} - \color{blue}{2} \]

      2. lower-/.f64 53.5

         \[\leadsto \frac{x}{y} - 2 \]

   4. Applied rewrites 53.5%

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

Alternative 11: 59.4% accurate, 1.7× speedup

Math

\[\begin{array}{l} t_1 := \frac{x}{y} - 2\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{-174}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{-135}:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (let* ((t_1 (- (/ x y) 2.0)))
   (if (<= t -9.5e-174) t_1 (if (<= t 9e-135) (/ 2.0 t) t_1))))

C

double code(double x, double y, double z, double t) {
	double t_1 = (x / y) - 2.0;
	double tmp;
	if (t <= -9.5e-174) {
		tmp = t_1;
	} else if (t <= 9e-135) {
		tmp = 2.0 / t;
	} else {
		tmp = t_1;
	}
	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)
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) :: t_1
    real(8) :: tmp
    t_1 = (x / y) - 2.0d0
    if (t <= (-9.5d-174)) then
        tmp = t_1
    else if (t <= 9d-135) then
        tmp = 2.0d0 / t
    else
        tmp = t_1
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double t_1 = (x / y) - 2.0;
	double tmp;
	if (t <= -9.5e-174) {
		tmp = t_1;
	} else if (t <= 9e-135) {
		tmp = 2.0 / t;
	} else {
		tmp = t_1;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	t_1 = (x / y) - 2.0
	tmp = 0
	if t <= -9.5e-174:
		tmp = t_1
	elif t <= 9e-135:
		tmp = 2.0 / t
	else:
		tmp = t_1
	return tmp

Julia

function code(x, y, z, t)
	t_1 = Float64(Float64(x / y) - 2.0)
	tmp = 0.0
	if (t <= -9.5e-174)
		tmp = t_1;
	elseif (t <= 9e-135)
		tmp = Float64(2.0 / t);
	else
		tmp = t_1;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	t_1 = (x / y) - 2.0;
	tmp = 0.0;
	if (t <= -9.5e-174)
		tmp = t_1;
	elseif (t <= 9e-135)
		tmp = 2.0 / t;
	else
		tmp = t_1;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := Block[{t$95$1 = N[(N[(x / y), $MachinePrecision] - 2.0), $MachinePrecision]}, If[LessEqual[t, -9.5e-174], t$95$1, If[LessEqual[t, 9e-135], N[(2.0 / t), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.50000000000000075e-174 or 8.99999999999999975e-135 < t

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x}{y} - \color{blue}{2} \]

      2. lower-/.f64 53.5

         \[\leadsto \frac{x}{y} - 2 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]

   if -9.50000000000000075e-174 < t < 8.99999999999999975e-135

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      3. lower-+.f64 86.5

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} + \frac{x}{y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{z \cdot t}} + \frac{x}{y} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}}{z}}{t} + \frac{x}{y} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right) + 2}}{z}}{t} + \frac{x}{y} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right)} + 2}{z}}{t} + \frac{x}{y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(1 - t\right) + 2}{z}}{t} + \frac{x}{y} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\color{blue}{z \cdot \left(2 \cdot \left(1 - t\right)\right)} + 2}{z}}{t} + \frac{x}{y} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \left(1 - t\right)\right) \cdot z} + 2}{z}}{t} + \frac{x}{y} \]

      15. add-to-fraction-rev N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left(1 - t\right) + \frac{2}{z}}}{t} + \frac{x}{y} \]

      16. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 - t\right) \cdot 2} + \frac{2}{z}}{t} + \frac{x}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}}{t} + \frac{x}{y} \]

      18. lower-/.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, 2, \color{blue}{\frac{2}{z}}\right)}{t} + \frac{x}{y} \]

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t} + \frac{x}{y}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}} + \frac{x}{y} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right) \cdot \frac{1}{t}} + \frac{x}{y} \]

      3. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - t\right) \cdot 2 + \frac{2}{z}\right)} \cdot \frac{1}{t} + \frac{x}{y} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(1 - t\right) \cdot 2 + \color{blue}{\frac{2}{z}}\right) \cdot \frac{1}{t} + \frac{x}{y} \]

      5. add-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2}{z}} \cdot \frac{1}{t} + \frac{x}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2\right) \cdot \frac{1}{t}}{z}} + \frac{x}{y} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 - t\right) \cdot \left(2 \cdot z\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(1 - t\right) \cdot \color{blue}{\left(z \cdot 2\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, z \cdot 2, 2\right)} \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{2 \cdot z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      11. count-2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      13. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}}{z} + \frac{x}{y} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}{z}} + \frac{x}{y} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(z + z, 1 - t, 2\right)}{t}}{z}} + \frac{x}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{\color{blue}{t}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      4. lower-/.f64 48.3

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]

   9. Taylor expanded in z around inf

      \[\leadsto \frac{2}{t} \]
   10. Step-by-step derivation

   11. Applied rewrites 20.0%

       \[\leadsto \frac{2}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 12: 37.2% accurate, 2.1× speedup

Math

\[\begin{array}{l} \mathbf{if}\;t \leq -0.031:\\ \;\;\;\;-2\\ \mathbf{elif}\;t \leq 110000000:\\ \;\;\;\;\frac{2}{t}\\ \mathbf{else}:\\ \;\;\;\;-2\\ \end{array} \]

FPCore

(FPCore (x y z t)
 :precision binary64
 (if (<= t -0.031) -2.0 (if (<= t 110000000.0) (/ 2.0 t) -2.0)))

C

double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -0.031) {
		tmp = -2.0;
	} else if (t <= 110000000.0) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	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)
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) :: tmp
    if (t <= (-0.031d0)) then
        tmp = -2.0d0
    else if (t <= 110000000.0d0) then
        tmp = 2.0d0 / t
    else
        tmp = -2.0d0
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t) {
	double tmp;
	if (t <= -0.031) {
		tmp = -2.0;
	} else if (t <= 110000000.0) {
		tmp = 2.0 / t;
	} else {
		tmp = -2.0;
	}
	return tmp;
}

Python

def code(x, y, z, t):
	tmp = 0
	if t <= -0.031:
		tmp = -2.0
	elif t <= 110000000.0:
		tmp = 2.0 / t
	else:
		tmp = -2.0
	return tmp

Julia

function code(x, y, z, t)
	tmp = 0.0
	if (t <= -0.031)
		tmp = -2.0;
	elseif (t <= 110000000.0)
		tmp = Float64(2.0 / t);
	else
		tmp = -2.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t)
	tmp = 0.0;
	if (t <= -0.031)
		tmp = -2.0;
	elseif (t <= 110000000.0)
		tmp = 2.0 / t;
	else
		tmp = -2.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_] := If[LessEqual[t, -0.031], -2.0, If[LessEqual[t, 110000000.0], N[(2.0 / t), $MachinePrecision], -2.0]]

Derivation

1. Split input into 2 regimes

2. if t < -0.031 or 1.1e8 < t

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

   2. Taylor expanded in t around inf

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]
   3. Step-by-step derivation

      1. lower--.f64 N/A

         \[\leadsto \frac{x}{y} - \color{blue}{2} \]

      2. lower-/.f64 53.5

         \[\leadsto \frac{x}{y} - 2 \]

   4. Applied rewrites 53.5%

      \[\leadsto \color{blue}{\frac{x}{y} - 2} \]

   5. Taylor expanded in x around 0

      \[\leadsto -2 \]
   6. Step-by-step derivation

   7. Applied rewrites 20.0%

      \[\leadsto -2 \]

   if -0.031 < t < 1.1e8

   1. Initial program 86.5%

      \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]
   2. Step-by-step derivation

      1. lift-+.f64 N/A

         \[\leadsto \color{blue}{\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} \]

      2. +-commutative N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      3. lower-+.f64 86.5

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} + \frac{x}{y}} \]

      4. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z}} + \frac{x}{y} \]

      5. lift-*.f64 N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{t \cdot z}} + \frac{x}{y} \]

      6. *-commutative N/A

         \[\leadsto \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{\color{blue}{z \cdot t}} + \frac{x}{y} \]

      7. associate-/r* N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      8. lower-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{z}}{t}} + \frac{x}{y} \]

      9. lift-+.f64 N/A

         \[\leadsto \frac{\frac{\color{blue}{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}}{z}}{t} + \frac{x}{y} \]

      10. +-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right) + 2}}{z}}{t} + \frac{x}{y} \]

      11. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right) \cdot \left(1 - t\right)} + 2}{z}}{t} + \frac{x}{y} \]

      12. lift-*.f64 N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(z \cdot 2\right)} \cdot \left(1 - t\right) + 2}{z}}{t} + \frac{x}{y} \]

      13. associate-*l* N/A

          \[\leadsto \frac{\frac{\color{blue}{z \cdot \left(2 \cdot \left(1 - t\right)\right)} + 2}{z}}{t} + \frac{x}{y} \]

      14. *-commutative N/A

          \[\leadsto \frac{\frac{\color{blue}{\left(2 \cdot \left(1 - t\right)\right) \cdot z} + 2}{z}}{t} + \frac{x}{y} \]

      15. add-to-fraction-rev N/A

          \[\leadsto \frac{\color{blue}{2 \cdot \left(1 - t\right) + \frac{2}{z}}}{t} + \frac{x}{y} \]

      16. *-commutative N/A

          \[\leadsto \frac{\color{blue}{\left(1 - t\right) \cdot 2} + \frac{2}{z}}{t} + \frac{x}{y} \]

      17. lower-fma.f64 N/A

          \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}}{t} + \frac{x}{y} \]

      18. lower-/.f64 99.2

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, 2, \color{blue}{\frac{2}{z}}\right)}{t} + \frac{x}{y} \]

   3. Applied rewrites 99.2%

      \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t} + \frac{x}{y}} \]
   4. Step-by-step derivation

      1. lift-/.f64 N/A

         \[\leadsto \color{blue}{\frac{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right)}{t}} + \frac{x}{y} \]

      2. mult-flip N/A

         \[\leadsto \color{blue}{\mathsf{fma}\left(1 - t, 2, \frac{2}{z}\right) \cdot \frac{1}{t}} + \frac{x}{y} \]

      3. lift-fma.f64 N/A

         \[\leadsto \color{blue}{\left(\left(1 - t\right) \cdot 2 + \frac{2}{z}\right)} \cdot \frac{1}{t} + \frac{x}{y} \]

      4. lift-/.f64 N/A

         \[\leadsto \left(\left(1 - t\right) \cdot 2 + \color{blue}{\frac{2}{z}}\right) \cdot \frac{1}{t} + \frac{x}{y} \]

      5. add-to-fraction N/A

         \[\leadsto \color{blue}{\frac{\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2}{z}} \cdot \frac{1}{t} + \frac{x}{y} \]

      6. associate-*l/ N/A

         \[\leadsto \color{blue}{\frac{\left(\left(\left(1 - t\right) \cdot 2\right) \cdot z + 2\right) \cdot \frac{1}{t}}{z}} + \frac{x}{y} \]

      7. associate-*l* N/A

         \[\leadsto \frac{\left(\color{blue}{\left(1 - t\right) \cdot \left(2 \cdot z\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      8. *-commutative N/A

         \[\leadsto \frac{\left(\left(1 - t\right) \cdot \color{blue}{\left(z \cdot 2\right)} + 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      9. lower-fma.f64 N/A

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left(1 - t, z \cdot 2, 2\right)} \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      10. *-commutative N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{2 \cdot z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      11. count-2 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      12. lift-+.f64 N/A

          \[\leadsto \frac{\mathsf{fma}\left(1 - t, \color{blue}{z + z}, 2\right) \cdot \frac{1}{t}}{z} + \frac{x}{y} \]

      13. mult-flip N/A

          \[\leadsto \frac{\color{blue}{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}}{z} + \frac{x}{y} \]

      14. lower-/.f64 N/A

          \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(1 - t, z + z, 2\right)}{t}}{z}} + \frac{x}{y} \]

   5. Applied rewrites 76.5%

      \[\leadsto \color{blue}{\frac{\frac{\mathsf{fma}\left(z + z, 1 - t, 2\right)}{t}}{z}} + \frac{x}{y} \]

   6. Taylor expanded in t around 0

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]
   7. Step-by-step derivation

      1. lower-/.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{\color{blue}{t}} \]

      2. lower-+.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      3. lower-*.f64 N/A

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

      4. lower-/.f64 48.3

         \[\leadsto \frac{2 + 2 \cdot \frac{1}{z}}{t} \]

   8. Applied rewrites 48.3%

      \[\leadsto \color{blue}{\frac{2 + 2 \cdot \frac{1}{z}}{t}} \]

   9. Taylor expanded in z around inf

      \[\leadsto \frac{2}{t} \]
   10. Step-by-step derivation

   11. Applied rewrites 20.0%

       \[\leadsto \frac{2}{t} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 13: 20.0% accurate, 24.7× speedup

Math

\[-2 \]

FPCore

(FPCore (x y z t) :precision binary64 -2.0)

C

double code(double x, double y, double z, double t) {
	return -2.0;
}

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)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    code = -2.0d0
end function

Java

public static double code(double x, double y, double z, double t) {
	return -2.0;
}

Python

def code(x, y, z, t):
	return -2.0

Julia

function code(x, y, z, t)
	return -2.0
end

MATLAB

function tmp = code(x, y, z, t)
	tmp = -2.0;
end

Wolfram

code[x_, y_, z_, t_] := -2.0

Derivation

1. Initial program 86.5%

   \[\frac{x}{y} + \frac{2 + \left(z \cdot 2\right) \cdot \left(1 - t\right)}{t \cdot z} \]

2. Taylor expanded in t around inf

   \[\leadsto \color{blue}{\frac{x}{y} - 2} \]
3. Step-by-step derivation

   1. lower--.f64 N/A

      \[\leadsto \frac{x}{y} - \color{blue}{2} \]

   2. lower-/.f64 53.5

      \[\leadsto \frac{x}{y} - 2 \]

4. Applied rewrites 53.5%

   \[\leadsto \color{blue}{\frac{x}{y} - 2} \]

5. Taylor expanded in x around 0

   \[\leadsto -2 \]
6. Step-by-step derivation

7. Applied rewrites 20.0%

   \[\leadsto -2 \]
8. Add Preprocessing

Reproduce

herbie shell --seed 2025172 
(FPCore (x y z t)
  :name "Data.HashTable.ST.Basic:computeOverhead from hashtables-1.2.0.2"
  :precision binary64
  (+ (/ x y) (/ (+ 2.0 (* (* z 2.0) (- 1.0 t))) (* t z))))