Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3

Percentage Accurate: 96.9% → 99.7%

Time: 7.0s

Alternatives: 13

Speedup: 1.1×

Specification

Math

\[\begin{array}{l} \\ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 96.9% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ x - \frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    code = x - ((y - z) / (((t - z) + 1.0d0) / a))
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Alternative 1: 99.7% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \mathsf{fma}\left(\frac{z - y}{\left(t - z\right) - -1}, a, x\right) \end{array} \]

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

   1. lift--.f64 N/A

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

   2. sub-flip N/A

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

   3. +-commutative N/A

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

   4. lift-/.f64 N/A

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

   5. frac-2neg N/A

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

   6. distribute-neg-frac2 N/A

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

   7. remove-double-neg N/A

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

   8. lift-/.f64 N/A

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

   9. associate-/r/ N/A

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

   10. lower-fma.f64 N/A

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

   11. lower-/.f64 N/A

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

   12. lift--.f64 N/A

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

   13. sub-negate-rev N/A

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

   14. lower--.f64 99.7

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

   15. lift-+.f64 N/A

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

   16. add-flip N/A

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

   17. lower--.f64 N/A

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

   18. metadata-eval 99.7

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

3. Applied rewrites 99.7%

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

4. Taylor expanded in y around 0

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

6. Applied rewrites 74.7%

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

7. Taylor expanded in z around 0

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

   1. lower--.f64 99.7

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

9. Applied rewrites 99.7%

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

Alternative 2: 97.6% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \left(t - z\right) - -1\\ \mathbf{if}\;z \leq -7 \cdot 10^{+197}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t\_1}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{a}{t\_1}, z - y, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- (- t z) -1.0)))
   (if (<= z -7e+197) (fma (/ z t_1) a x) (fma (/ a t_1) (- z y) x))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = (t - z) - -1.0;
	double tmp;
	if (z <= -7e+197) {
		tmp = fma((z / t_1), a, x);
	} else {
		tmp = fma((a / t_1), (z - y), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(Float64(t - z) - -1.0)
	tmp = 0.0
	if (z <= -7e+197)
		tmp = fma(Float64(z / t_1), a, x);
	else
		tmp = fma(Float64(a / t_1), Float64(z - y), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision]}, If[LessEqual[z, -7e+197], N[(N[(z / t$95$1), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(a / t$95$1), $MachinePrecision] * N[(z - y), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -6.99999999999999999e197

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 74.7%

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

   if -6.99999999999999999e197 < z

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. mult-flip N/A

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

      6. *-commutative N/A

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

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

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

      8. lower-fma.f64 N/A

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

      9. lift-/.f64 N/A

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

      10. div-flip-rev N/A

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

      11. lower-/.f64 N/A

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

      12. lift-+.f64 N/A

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

      13. add-flip N/A

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

      14. lower--.f64 N/A

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

      15. metadata-eval N/A

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

      16. lift--.f64 N/A

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

      17. sub-negate-rev N/A

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

      18. lower--.f64 97.2

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

   3. Applied rewrites 97.2%

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

Alternative 3: 91.9% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{if}\;t \leq -4.2 \cdot 10^{+38}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 2.65 \cdot 10^{+57}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{1 - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) t) a x)))
   (if (<= t -4.2e+38)
     t_1
     (if (<= t 2.65e+57) (fma (/ (- z y) (- 1.0 z)) a x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / t), a, x);
	double tmp;
	if (t <= -4.2e+38) {
		tmp = t_1;
	} else if (t <= 2.65e+57) {
		tmp = fma(((z - y) / (1.0 - z)), a, x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / t), a, x)
	tmp = 0.0
	if (t <= -4.2e+38)
		tmp = t_1;
	elseif (t <= 2.65e+57)
		tmp = fma(Float64(Float64(z - y) / Float64(1.0 - z)), a, x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[t, -4.2e+38], t$95$1, If[LessEqual[t, 2.65e+57], N[(N[(N[(z - y), $MachinePrecision] / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -4.2e38 or 2.64999999999999993e57 < t

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 53.1

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

   6. Applied rewrites 53.1%

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

   if -4.2e38 < t < 2.64999999999999993e57

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around 0

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

      1. lower--.f64 80.7

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

   6. Applied rewrites 80.7%

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

Alternative 4: 88.5% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z}{\left(t - z\right) - -1}, a, x\right)\\ \mathbf{if}\;z \leq -1.8 \cdot 10^{-5}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;z \leq 68000000000:\\ \;\;\;\;x - \frac{a}{t - -1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ z (- (- t z) -1.0)) a x)))
   (if (<= z -1.8e-5)
     t_1
     (if (<= z 68000000000.0) (- x (* (/ a (- t -1.0)) y)) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma((z / ((t - z) - -1.0)), a, x);
	double tmp;
	if (z <= -1.8e-5) {
		tmp = t_1;
	} else if (z <= 68000000000.0) {
		tmp = x - ((a / (t - -1.0)) * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(z / Float64(Float64(t - z) - -1.0)), a, x)
	tmp = 0.0
	if (z <= -1.8e-5)
		tmp = t_1;
	elseif (z <= 68000000000.0)
		tmp = Float64(x - Float64(Float64(a / Float64(t - -1.0)) * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(z / N[(N[(t - z), $MachinePrecision] - -1.0), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[z, -1.8e-5], t$95$1, If[LessEqual[z, 68000000000.0], N[(x - N[(N[(a / N[(t - -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if z < -1.80000000000000005e-5 or 6.8e10 < z

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 74.7%

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

   if -1.80000000000000005e-5 < z < 6.8e10

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.8

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

   4. Applied rewrites 69.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

         \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      9. lower-/.f64 71.8

         \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      10. lift-+.f64 N/A

          \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      11. +-commutative N/A

          \[\leadsto x - \frac{a}{t + 1} \cdot y \]

      12. add-flip N/A

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

      13. metadata-eval N/A

          \[\leadsto x - \frac{a}{t - -1} \cdot y \]

      14. lower--.f64 71.8

          \[\leadsto x - \frac{a}{t - -1} \cdot y \]

   6. Applied rewrites 71.8%

      \[\leadsto \color{blue}{x - \frac{a}{t - -1} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 5: 85.6% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+78}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 68000000000:\\ \;\;\;\;x - \frac{a}{t - -1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{\left(t - -1\right) - z}, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.5e+78)
   (- x a)
   (if (<= z 68000000000.0)
     (- x (* (/ a (- t -1.0)) y))
     (fma z (/ a (- (- t -1.0) z)) x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.5e+78) {
		tmp = x - a;
	} else if (z <= 68000000000.0) {
		tmp = x - ((a / (t - -1.0)) * y);
	} else {
		tmp = fma(z, (a / ((t - -1.0) - z)), x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.5e+78)
		tmp = Float64(x - a);
	elseif (z <= 68000000000.0)
		tmp = Float64(x - Float64(Float64(a / Float64(t - -1.0)) * y));
	else
		tmp = fma(z, Float64(a / Float64(Float64(t - -1.0) - z)), x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.5e+78], N[(x - a), $MachinePrecision], If[LessEqual[z, 68000000000.0], N[(x - N[(N[(a / N[(t - -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(z * N[(a / N[(N[(t - -1.0), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -3.5000000000000001e78

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 60.5%

      \[\leadsto x - \color{blue}{a} \]

   if -3.5000000000000001e78 < z < 6.8e10

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.8

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

   4. Applied rewrites 69.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

         \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      9. lower-/.f64 71.8

         \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      10. lift-+.f64 N/A

          \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      11. +-commutative N/A

          \[\leadsto x - \frac{a}{t + 1} \cdot y \]

      12. add-flip N/A

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

      13. metadata-eval N/A

          \[\leadsto x - \frac{a}{t - -1} \cdot y \]

      14. lower--.f64 71.8

          \[\leadsto x - \frac{a}{t - -1} \cdot y \]

   6. Applied rewrites 71.8%

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

   if 6.8e10 < z

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 74.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. metadata-eval N/A

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

      9. add-flip N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 73.0%

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

Alternative 6: 84.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.5 \cdot 10^{+78}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.12 \cdot 10^{+90}:\\ \;\;\;\;x - \frac{a}{t - -1} \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.5e+78)
   (- x a)
   (if (<= z 1.12e+90) (- x (* (/ a (- t -1.0)) y)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.5e+78) {
		tmp = x - a;
	} else if (z <= 1.12e+90) {
		tmp = x - ((a / (t - -1.0)) * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-3.5d+78)) then
        tmp = x - a
    else if (z <= 1.12d+90) then
        tmp = x - ((a / (t - (-1.0d0))) * y)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.5e+78) {
		tmp = x - a;
	} else if (z <= 1.12e+90) {
		tmp = x - ((a / (t - -1.0)) * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.5e+78:
		tmp = x - a
	elif z <= 1.12e+90:
		tmp = x - ((a / (t - -1.0)) * y)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.5e+78)
		tmp = Float64(x - a);
	elseif (z <= 1.12e+90)
		tmp = Float64(x - Float64(Float64(a / Float64(t - -1.0)) * y));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.5e+78)
		tmp = x - a;
	elseif (z <= 1.12e+90)
		tmp = x - ((a / (t - -1.0)) * y);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.5e+78], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.12e+90], N[(x - N[(N[(a / N[(t - -1.0), $MachinePrecision]), $MachinePrecision] * y), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.5000000000000001e78 or 1.1199999999999999e90 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 60.5%

      \[\leadsto x - \color{blue}{a} \]

   if -3.5000000000000001e78 < z < 1.1199999999999999e90

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.8

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

   4. Applied rewrites 69.8%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. *-commutative N/A

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

      7. lower-*.f64 N/A

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

      8. mult-flip-rev N/A

         \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      9. lower-/.f64 71.8

         \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      10. lift-+.f64 N/A

          \[\leadsto x - \frac{a}{1 + t} \cdot y \]

      11. +-commutative N/A

          \[\leadsto x - \frac{a}{t + 1} \cdot y \]

      12. add-flip N/A

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

      13. metadata-eval N/A

          \[\leadsto x - \frac{a}{t - -1} \cdot y \]

      14. lower--.f64 71.8

          \[\leadsto x - \frac{a}{t - -1} \cdot y \]

   6. Applied rewrites 71.8%

      \[\leadsto \color{blue}{x - \frac{a}{t - -1} \cdot y} \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 7: 79.3% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{if}\;t \leq -9.8 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq -1.8 \cdot 10^{-70}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{1 - z}, x\right)\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+57}:\\ \;\;\;\;x - \frac{y}{1 - z} \cdot a\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) t) a x)))
   (if (<= t -9.8e+30)
     t_1
     (if (<= t -1.8e-70)
       (fma z (/ a (- 1.0 z)) x)
       (if (<= t 2.25e+57) (- x (* (/ y (- 1.0 z)) a)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / t), a, x);
	double tmp;
	if (t <= -9.8e+30) {
		tmp = t_1;
	} else if (t <= -1.8e-70) {
		tmp = fma(z, (a / (1.0 - z)), x);
	} else if (t <= 2.25e+57) {
		tmp = x - ((y / (1.0 - z)) * a);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / t), a, x)
	tmp = 0.0
	if (t <= -9.8e+30)
		tmp = t_1;
	elseif (t <= -1.8e-70)
		tmp = fma(z, Float64(a / Float64(1.0 - z)), x);
	elseif (t <= 2.25e+57)
		tmp = Float64(x - Float64(Float64(y / Float64(1.0 - z)) * a));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[t, -9.8e+30], t$95$1, If[LessEqual[t, -1.8e-70], N[(z * N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 2.25e+57], N[(x - N[(N[(y / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.79999999999999969e30 or 2.24999999999999998e57 < t

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 53.1

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

   6. Applied rewrites 53.1%

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

   if -9.79999999999999969e30 < t < -1.8000000000000001e-70

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 74.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. metadata-eval N/A

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

      9. add-flip N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 73.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 66.0

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

   11. Applied rewrites 66.0%

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

   if -1.8000000000000001e-70 < t < 2.24999999999999998e57

   1. Initial program 96.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 71.2

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      3. lower--.f64 64.1

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

   7. Applied rewrites 64.1%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      6. lower-*.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      7. lower-/.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      8. lift--.f64 65.7

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

   9. Applied rewrites 65.7%

      \[\leadsto x - \color{blue}{\frac{y}{1 - z} \cdot a} \]
3. Recombined 3 regimes into one program.
4. Add Preprocessing

Alternative 8: 75.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := \mathsf{fma}\left(\frac{z - y}{t}, a, x\right)\\ \mathbf{if}\;t \leq -9.8 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 1.32 \cdot 10^{-197}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{1 - z}, x\right)\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+21}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (fma (/ (- z y) t) a x)))
   (if (<= t -9.8e+30)
     t_1
     (if (<= t 1.32e-197)
       (fma z (/ a (- 1.0 z)) x)
       (if (<= t 3e+21) (- x (* a y)) t_1)))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = fma(((z - y) / t), a, x);
	double tmp;
	if (t <= -9.8e+30) {
		tmp = t_1;
	} else if (t <= 1.32e-197) {
		tmp = fma(z, (a / (1.0 - z)), x);
	} else if (t <= 3e+21) {
		tmp = x - (a * y);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = fma(Float64(Float64(z - y) / t), a, x)
	tmp = 0.0
	if (t <= -9.8e+30)
		tmp = t_1;
	elseif (t <= 1.32e-197)
		tmp = fma(z, Float64(a / Float64(1.0 - z)), x);
	elseif (t <= 3e+21)
		tmp = Float64(x - Float64(a * y));
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(N[(N[(z - y), $MachinePrecision] / t), $MachinePrecision] * a + x), $MachinePrecision]}, If[LessEqual[t, -9.8e+30], t$95$1, If[LessEqual[t, 1.32e-197], N[(z * N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], If[LessEqual[t, 3e+21], N[(x - N[(a * y), $MachinePrecision]), $MachinePrecision], t$95$1]]]]

Derivation

1. Split input into 3 regimes

2. if t < -9.79999999999999969e30 or 3e21 < t

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower--.f64 53.1

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

   6. Applied rewrites 53.1%

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

   if -9.79999999999999969e30 < t < 1.32e-197

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 74.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. metadata-eval N/A

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

      9. add-flip N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 73.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 66.0

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

   11. Applied rewrites 66.0%

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

   if 1.32e-197 < t < 3e21

   1. Initial program 96.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 71.2

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      3. lower--.f64 64.1

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

   7. Applied rewrites 64.1%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      6. lower-*.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      7. lower-/.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      8. lift--.f64 65.7

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

   9. Applied rewrites 65.7%

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

   10. Taylor expanded in z around 0

       \[\leadsto x - a \cdot \color{blue}{y} \]
   11. Step-by-step derivation

       1. lower-*.f64 58.2

          \[\leadsto x - a \cdot y \]

   12. Applied rewrites 58.2%

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

Alternative 9: 73.4% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.2 \cdot 10^{-20}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-257}:\\ \;\;\;\;x - \frac{y}{t} \cdot a\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-5}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -2.2e-20)
   (- x a)
   (if (<= z -1.45e-257)
     (- x (* (/ y t) a))
     (if (<= z 1.52e-5) (- x (* a y)) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-20) {
		tmp = x - a;
	} else if (z <= -1.45e-257) {
		tmp = x - ((y / t) * a);
	} else if (z <= 1.52e-5) {
		tmp = x - (a * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-2.2d-20)) then
        tmp = x - a
    else if (z <= (-1.45d-257)) then
        tmp = x - ((y / t) * a)
    else if (z <= 1.52d-5) then
        tmp = x - (a * y)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -2.2e-20) {
		tmp = x - a;
	} else if (z <= -1.45e-257) {
		tmp = x - ((y / t) * a);
	} else if (z <= 1.52e-5) {
		tmp = x - (a * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -2.2e-20:
		tmp = x - a
	elif z <= -1.45e-257:
		tmp = x - ((y / t) * a)
	elif z <= 1.52e-5:
		tmp = x - (a * y)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -2.2e-20)
		tmp = Float64(x - a);
	elseif (z <= -1.45e-257)
		tmp = Float64(x - Float64(Float64(y / t) * a));
	elseif (z <= 1.52e-5)
		tmp = Float64(x - Float64(a * y));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -2.2e-20)
		tmp = x - a;
	elseif (z <= -1.45e-257)
		tmp = x - ((y / t) * a);
	elseif (z <= 1.52e-5)
		tmp = x - (a * y);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -2.2e-20], N[(x - a), $MachinePrecision], If[LessEqual[z, -1.45e-257], N[(x - N[(N[(y / t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.52e-5], N[(x - N[(a * y), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -2.19999999999999991e-20 or 1.52e-5 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 60.5%

      \[\leadsto x - \color{blue}{a} \]

   if -2.19999999999999991e-20 < z < -1.4500000000000001e-257

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.9

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

   9. Applied rewrites 55.9%

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

   if -1.4500000000000001e-257 < z < 1.52e-5

   1. Initial program 96.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 71.2

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      3. lower--.f64 64.1

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

   7. Applied rewrites 64.1%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      6. lower-*.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      7. lower-/.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      8. lift--.f64 65.7

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

   9. Applied rewrites 65.7%

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

   10. Taylor expanded in z around 0

       \[\leadsto x - a \cdot \color{blue}{y} \]
   11. Step-by-step derivation

       1. lower-*.f64 58.2

          \[\leadsto x - a \cdot y \]

   12. Applied rewrites 58.2%

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

Alternative 10: 72.5% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1 \cdot 10^{+60}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq -1.45 \cdot 10^{-257}:\\ \;\;\;\;x - y \cdot \frac{a}{t}\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-5}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -1e+60)
   (- x a)
   (if (<= z -1.45e-257)
     (- x (* y (/ a t)))
     (if (<= z 1.52e-5) (- x (* a y)) (- x a)))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+60) {
		tmp = x - a;
	} else if (z <= -1.45e-257) {
		tmp = x - (y * (a / t));
	} else if (z <= 1.52e-5) {
		tmp = x - (a * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-1d+60)) then
        tmp = x - a
    else if (z <= (-1.45d-257)) then
        tmp = x - (y * (a / t))
    else if (z <= 1.52d-5) then
        tmp = x - (a * y)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -1e+60) {
		tmp = x - a;
	} else if (z <= -1.45e-257) {
		tmp = x - (y * (a / t));
	} else if (z <= 1.52e-5) {
		tmp = x - (a * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -1e+60:
		tmp = x - a
	elif z <= -1.45e-257:
		tmp = x - (y * (a / t))
	elif z <= 1.52e-5:
		tmp = x - (a * y)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -1e+60)
		tmp = Float64(x - a);
	elseif (z <= -1.45e-257)
		tmp = Float64(x - Float64(y * Float64(a / t)));
	elseif (z <= 1.52e-5)
		tmp = Float64(x - Float64(a * y));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -1e+60)
		tmp = x - a;
	elseif (z <= -1.45e-257)
		tmp = x - (y * (a / t));
	elseif (z <= 1.52e-5)
		tmp = x - (a * y);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -1e+60], N[(x - a), $MachinePrecision], If[LessEqual[z, -1.45e-257], N[(x - N[(y * N[(a / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[z, 1.52e-5], N[(x - N[(a * y), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]]

Derivation

1. Split input into 3 regimes

2. if z < -9.9999999999999995e59 or 1.52e-5 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 60.5%

      \[\leadsto x - \color{blue}{a} \]

   if -9.9999999999999995e59 < z < -1.4500000000000001e-257

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

      1. lift-/.f64 N/A

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

      2. mult-flip N/A

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

      3. lift-*.f64 N/A

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

      4. *-commutative N/A

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

      5. associate-*l* N/A

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

      6. lower-*.f64 N/A

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

      7. mult-flip-rev N/A

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

      8. lower-/.f64 55.5

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

   9. Applied rewrites 55.5%

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

   if -1.4500000000000001e-257 < z < 1.52e-5

   1. Initial program 96.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 71.2

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      3. lower--.f64 64.1

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

   7. Applied rewrites 64.1%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      6. lower-*.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      7. lower-/.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      8. lift--.f64 65.7

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

   9. Applied rewrites 65.7%

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

   10. Taylor expanded in z around 0

       \[\leadsto x - a \cdot \color{blue}{y} \]
   11. Step-by-step derivation

       1. lower-*.f64 58.2

          \[\leadsto x - a \cdot y \]

   12. Applied rewrites 58.2%

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

Alternative 11: 72.3% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_1 := x - \frac{y}{t} \cdot a\\ \mathbf{if}\;t \leq -9.5 \cdot 10^{+30}:\\ \;\;\;\;t\_1\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-77}:\\ \;\;\;\;\mathsf{fma}\left(z, \frac{a}{1 - z}, x\right)\\ \mathbf{else}:\\ \;\;\;\;t\_1\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (let* ((t_1 (- x (* (/ y t) a))))
   (if (<= t -9.5e+30) t_1 (if (<= t 3.5e-77) (fma z (/ a (- 1.0 z)) x) t_1))))

C

double code(double x, double y, double z, double t, double a) {
	double t_1 = x - ((y / t) * a);
	double tmp;
	if (t <= -9.5e+30) {
		tmp = t_1;
	} else if (t <= 3.5e-77) {
		tmp = fma(z, (a / (1.0 - z)), x);
	} else {
		tmp = t_1;
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	t_1 = Float64(x - Float64(Float64(y / t) * a))
	tmp = 0.0
	if (t <= -9.5e+30)
		tmp = t_1;
	elseif (t <= 3.5e-77)
		tmp = fma(z, Float64(a / Float64(1.0 - z)), x);
	else
		tmp = t_1;
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := Block[{t$95$1 = N[(x - N[(N[(y / t), $MachinePrecision] * a), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -9.5e+30], t$95$1, If[LessEqual[t, 3.5e-77], N[(z * N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision] + x), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -9.5000000000000003e30 or 3.50000000000000013e-77 < t

   1. Initial program 96.9%

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

   2. Taylor expanded in z around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower-+.f64 69.8

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

   4. Applied rewrites 69.8%

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

   5. Taylor expanded in t around inf

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 54.3

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

   7. Applied rewrites 54.3%

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

      1. lift-/.f64 N/A

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

      2. lift-*.f64 N/A

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

      3. associate-/l* N/A

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

      4. *-commutative N/A

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

      5. lower-*.f64 N/A

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

      6. lower-/.f64 55.9

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

   9. Applied rewrites 55.9%

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

   if -9.5000000000000003e30 < t < 3.50000000000000013e-77

   1. Initial program 96.9%

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

      1. lift--.f64 N/A

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

      2. sub-flip N/A

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

      3. +-commutative N/A

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

      4. lift-/.f64 N/A

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

      5. frac-2neg N/A

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

      6. distribute-neg-frac2 N/A

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

      7. remove-double-neg N/A

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

      8. lift-/.f64 N/A

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

      9. associate-/r/ N/A

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

      10. lower-fma.f64 N/A

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

      11. lower-/.f64 N/A

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

      12. lift--.f64 N/A

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

      13. sub-negate-rev N/A

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

      14. lower--.f64 99.7

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

      15. lift-+.f64 N/A

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

      16. add-flip N/A

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

      17. lower--.f64 N/A

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

      18. metadata-eval 99.7

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

   3. Applied rewrites 99.7%

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

   4. Taylor expanded in y around 0

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

   6. Applied rewrites 74.7%

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

      1. lift-fma.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. associate-*l/ N/A

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

      4. associate-/l* N/A

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

      5. div-flip-rev N/A

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

      6. lift--.f64 N/A

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

      7. lift--.f64 N/A

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

      8. metadata-eval N/A

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

      9. add-flip N/A

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

      10. lower-fma.f64 N/A

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

   8. Applied rewrites 73.0%

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

   9. Taylor expanded in t around 0

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

       1. lower-/.f64 N/A

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

       2. lower--.f64 66.0

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

   11. Applied rewrites 66.0%

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

Alternative 12: 71.4% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -3.8 \cdot 10^{+77}:\\ \;\;\;\;x - a\\ \mathbf{elif}\;z \leq 1.52 \cdot 10^{-5}:\\ \;\;\;\;x - a \cdot y\\ \mathbf{else}:\\ \;\;\;\;x - a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -3.8e+77) (- x a) (if (<= z 1.52e-5) (- x (* a y)) (- x a))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.8e+77) {
		tmp = x - a;
	} else if (z <= 1.52e-5) {
		tmp = x - (a * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

real(8) function code(x, y, z, t, a)
use fmin_fmax_functions
    real(8), intent (in) :: x
    real(8), intent (in) :: y
    real(8), intent (in) :: z
    real(8), intent (in) :: t
    real(8), intent (in) :: a
    real(8) :: tmp
    if (z <= (-3.8d+77)) then
        tmp = x - a
    else if (z <= 1.52d-5) then
        tmp = x - (a * y)
    else
        tmp = x - a
    end if
    code = tmp
end function

Java

public static double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -3.8e+77) {
		tmp = x - a;
	} else if (z <= 1.52e-5) {
		tmp = x - (a * y);
	} else {
		tmp = x - a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if z <= -3.8e+77:
		tmp = x - a
	elif z <= 1.52e-5:
		tmp = x - (a * y)
	else:
		tmp = x - a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -3.8e+77)
		tmp = Float64(x - a);
	elseif (z <= 1.52e-5)
		tmp = Float64(x - Float64(a * y));
	else
		tmp = Float64(x - a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (z <= -3.8e+77)
		tmp = x - a;
	elseif (z <= 1.52e-5)
		tmp = x - (a * y);
	else
		tmp = x - a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -3.8e+77], N[(x - a), $MachinePrecision], If[LessEqual[z, 1.52e-5], N[(x - N[(a * y), $MachinePrecision]), $MachinePrecision], N[(x - a), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if z < -3.8000000000000001e77 or 1.52e-5 < z

   1. Initial program 96.9%

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

   2. Taylor expanded in z around inf

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

   4. Applied rewrites 60.5%

      \[\leadsto x - \color{blue}{a} \]

   if -3.8000000000000001e77 < z < 1.52e-5

   1. Initial program 96.9%

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

   2. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

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

      3. lower--.f64 N/A

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

      4. lower--.f64 71.2

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

   4. Applied rewrites 71.2%

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

   5. Taylor expanded in y around inf

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   6. Step-by-step derivation

      1. lower-/.f64 N/A

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

      2. lower-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      3. lower--.f64 64.1

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

   7. Applied rewrites 64.1%

      \[\leadsto x - \frac{a \cdot y}{\color{blue}{1 - z}} \]
   8. Step-by-step derivation

      1. lift--.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      2. lift-/.f64 N/A

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

      3. lift-*.f64 N/A

         \[\leadsto x - \frac{a \cdot y}{1 - z} \]

      4. associate-/l* N/A

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

      5. *-commutative N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      6. lower-*.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      7. lower-/.f64 N/A

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

      8. lift--.f64 65.7

         \[\leadsto x - \frac{y}{1 - z} \cdot a \]

   9. Applied rewrites 65.7%

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

   10. Taylor expanded in z around 0

       \[\leadsto x - a \cdot \color{blue}{y} \]
   11. Step-by-step derivation

       1. lower-*.f64 58.2

          \[\leadsto x - a \cdot y \]

   12. Applied rewrites 58.2%

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

Alternative 13: 60.5% accurate, 5.1× speedup

Math

\[\begin{array}{l} \\ x - a \end{array} \]

FPCore

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

C

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

Fortran

module fmin_fmax_functions
    implicit none
    private
    public fmax
    public fmin

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

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.9%

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

2. Taylor expanded in z around inf

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

4. Applied rewrites 60.5%

   \[\leadsto x - \color{blue}{a} \]
5. Add Preprocessing

Reproduce

herbie shell --seed 2025149 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64
  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))