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

Percentage Accurate: 97.1% → 99.8%

Time: 4.3s

Alternatives: 12

Speedup: 1.3×

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: 97.1% 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: 99.8% accurate, 1.3× speedup

Math

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

FPCore

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

C

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

Julia

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in a around 0

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

   1. +-commutative N/A

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

   2. *-commutative N/A

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

   3. lower-fma.f64 N/A

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

   4. sub-div N/A

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

   5. lower-/.f64 N/A

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

   6. lower--.f64 N/A

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

   7. lower--.f64 N/A

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

   8. lower-+.f64 99.9

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

5. Applied rewrites 99.9%

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

Alternative 2: 55.9% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\frac{y - z}{\frac{\left(t - z\right) + 1}{a}} \leq 2 \cdot 10^{+171}:\\ \;\;\;\;x\\ \mathbf{else}:\\ \;\;\;\;-a\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= (/ (- y z) (/ (+ (- t z) 1.0) a)) 2e+171) x (- a)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (((y - z) / (((t - z) + 1.0) / a)) <= 2e+171) {
		tmp = x;
	} else {
		tmp = -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 (((y - z) / (((t - z) + 1.0d0) / a)) <= 2d+171) then
        tmp = x
    else
        tmp = -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 (((y - z) / (((t - z) + 1.0) / a)) <= 2e+171) {
		tmp = x;
	} else {
		tmp = -a;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if ((y - z) / (((t - z) + 1.0) / a)) <= 2e+171:
		tmp = x
	else:
		tmp = -a
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (Float64(Float64(y - z) / Float64(Float64(Float64(t - z) + 1.0) / a)) <= 2e+171)
		tmp = x;
	else
		tmp = Float64(-a);
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if (((y - z) / (((t - z) + 1.0) / a)) <= 2e+171)
		tmp = x;
	else
		tmp = -a;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[N[(N[(y - z), $MachinePrecision] / N[(N[(N[(t - z), $MachinePrecision] + 1.0), $MachinePrecision] / a), $MachinePrecision]), $MachinePrecision], 2e+171], x, (-a)]

Derivation

1. Split input into 2 regimes

2. if (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a)) < 1.99999999999999991e171

   1. Initial program 95.6%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 55.8%

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

   if 1.99999999999999991e171 < (/.f64 (-.f64 y z) (/.f64 (+.f64 (-.f64 t z) #s(literal 1 binary64)) a))

   1. Initial program 99.7%

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

   3. Taylor expanded in a around inf

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

      1. *-commutative N/A

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

      2. lower-*.f64 N/A

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

      3. sub-div N/A

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

      4. lower-/.f64 N/A

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

      5. lower--.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower-+.f64 84.8

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

   5. Applied rewrites 84.8%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

         \[\leadsto \mathsf{neg}\left(a\right) \]

      2. lower-neg.f64 38.8

         \[\leadsto -a \]

   8. Applied rewrites 38.8%

      \[\leadsto -a \]
3. Recombined 2 regimes into one program.
4. Add Preprocessing

Alternative 3: 98.0% accurate, 0.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;t \leq -90000000000 \lor \neg \left(t \leq 1.02 \cdot 10^{-16}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{1 - z}\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= t -90000000000.0) (not (<= t 1.02e-16)))
   (fma (/ (- z y) (- t z)) a x)
   (- x (* (- y z) (/ a (- 1.0 z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((t <= -90000000000.0) || !(t <= 1.02e-16)) {
		tmp = fma(((z - y) / (t - z)), a, x);
	} else {
		tmp = x - ((y - z) * (a / (1.0 - z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((t <= -90000000000.0) || !(t <= 1.02e-16))
		tmp = fma(Float64(Float64(z - y) / Float64(t - z)), a, x);
	else
		tmp = Float64(x - Float64(Float64(y - z) * Float64(a / Float64(1.0 - z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[t, -90000000000.0], N[Not[LessEqual[t, 1.02e-16]], $MachinePrecision]], N[(N[(N[(z - y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - N[(N[(y - z), $MachinePrecision] * N[(a / N[(1.0 - z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if t < -9e10 or 1.0200000000000001e-16 < t

   1. Initial program 96.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 99.7%

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

   if -9e10 < t < 1.0200000000000001e-16

   1. Initial program 96.2%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 84.6

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

   5. Applied rewrites 84.6%

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

      1. lift--.f64 N/A

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

      2. lift-/.f64 N/A

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

      3. lift--.f64 N/A

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

      4. lift-*.f64 N/A

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

      5. associate-/l* N/A

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

      6. lower-*.f64 N/A

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

      7. lift--.f64 N/A

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

      8. lower-/.f64 N/A

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

      9. lift--.f64 95.4

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

   7. Applied rewrites 95.4%

      \[\leadsto \color{blue}{x - \left(y - z\right) \cdot \frac{a}{1 - z}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -90000000000 \lor \neg \left(t \leq 1.02 \cdot 10^{-16}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - \left(y - z\right) \cdot \frac{a}{1 - z}\\ \end{array} \]
5. Add Preprocessing

Alternative 4: 96.0% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-25} \lor \neg \left(z \leq 0.9\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.4e-25) (not (<= z 0.9)))
   (fma (/ (- z y) (- t z)) a x)
   (fma (/ y (- -1.0 t)) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.4e-25) || !(z <= 0.9)) {
		tmp = fma(((z - y) / (t - z)), a, x);
	} else {
		tmp = fma((y / (-1.0 - t)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.4e-25) || !(z <= 0.9))
		tmp = fma(Float64(Float64(z - y) / Float64(t - z)), a, x);
	else
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.4e-25], N[Not[LessEqual[z, 0.9]], $MachinePrecision]], N[(N[(N[(z - y), $MachinePrecision] / N[(t - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.39999999999999994e-25 or 0.900000000000000022 < z

   1. Initial program 93.2%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 97.3%

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

   if -1.39999999999999994e-25 < z < 0.900000000000000022

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-+.f64 91.1

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

   8. Applied rewrites 91.1%

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

4. Final simplification 94.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.4 \cdot 10^{-25} \lor \neg \left(z \leq 0.9\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 87.1% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-87} \lor \neg \left(z \leq 3 \cdot 10^{-69}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -7.6e-87) (not (<= z 3e-69)))
   (fma (/ z (- (+ 1.0 t) z)) a x)
   (fma (/ y (- -1.0 t)) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -7.6e-87) || !(z <= 3e-69)) {
		tmp = fma((z / ((1.0 + t) - z)), a, x);
	} else {
		tmp = fma((y / (-1.0 - t)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -7.6e-87) || !(z <= 3e-69))
		tmp = fma(Float64(z / Float64(Float64(1.0 + t) - z)), a, x);
	else
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -7.6e-87], N[Not[LessEqual[z, 3e-69]], $MachinePrecision]], N[(N[(z / N[(N[(1.0 + t), $MachinePrecision] - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -7.6e-87 or 2.99999999999999989e-69 < z

   1. Initial program 94.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in y around 0

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

   8. Applied rewrites 86.1%

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

   if -7.6e-87 < z < 2.99999999999999989e-69

   1. Initial program 98.9%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-+.f64 94.4

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

   8. Applied rewrites 94.4%

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

4. Final simplification 89.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -7.6 \cdot 10^{-87} \lor \neg \left(z \leq 3 \cdot 10^{-69}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{\left(1 + t\right) - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 88.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-23} \lor \neg \left(z \leq 2.9 \cdot 10^{+25}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1.05e-23) (not (<= z 2.9e+25)))
   (fma (/ z (- t z)) a x)
   (fma (/ y (- -1.0 t)) a x)))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1.05e-23) || !(z <= 2.9e+25)) {
		tmp = fma((z / (t - z)), a, x);
	} else {
		tmp = fma((y / (-1.0 - t)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1.05e-23) || !(z <= 2.9e+25))
		tmp = fma(Float64(z / Float64(t - z)), a, x);
	else
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1.05e-23], N[Not[LessEqual[z, 2.9e+25]], $MachinePrecision]], N[(N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.05e-23 or 2.8999999999999999e25 < z

   1. Initial program 93.0%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 98.0%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 85.4%

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

   if -1.05e-23 < z < 2.8999999999999999e25

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-+.f64 90.0

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

   8. Applied rewrites 90.0%

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

4. Final simplification 87.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1.05 \cdot 10^{-23} \lor \neg \left(z \leq 2.9 \cdot 10^{+25}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 78.2% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-38} \lor \neg \left(z \leq 3.3 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \left(y + y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -2.6e-38) (not (<= z 3.3e-6)))
   (fma (/ z (- t z)) a x)
   (- x (* a (+ y (* y z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -2.6e-38) || !(z <= 3.3e-6)) {
		tmp = fma((z / (t - z)), a, x);
	} else {
		tmp = x - (a * (y + (y * z)));
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -2.6e-38) || !(z <= 3.3e-6))
		tmp = fma(Float64(z / Float64(t - z)), a, x);
	else
		tmp = Float64(x - Float64(a * Float64(y + Float64(y * z))));
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -2.6e-38], N[Not[LessEqual[z, 3.3e-6]], $MachinePrecision]], N[(N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(x - N[(a * N[(y + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -2.60000000000000011e-38 or 3.30000000000000017e-6 < z

   1. Initial program 93.3%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 97.3%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 83.1%

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

   if -2.60000000000000011e-38 < z < 3.30000000000000017e-6

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 76.1

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

   5. Applied rewrites 76.1%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 71.0

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

   8. Applied rewrites 71.0%

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

   9. Taylor expanded in z around 0

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

       1. lower-+.f64 N/A

          \[\leadsto x - a \cdot \left(y + y \cdot z\right) \]

       2. lower-*.f64 71.0

          \[\leadsto x - a \cdot \left(y + y \cdot z\right) \]

   11. Applied rewrites 71.0%

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

4. Final simplification 77.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -2.6 \cdot 10^{-38} \lor \neg \left(z \leq 3.3 \cdot 10^{-6}\right):\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - z}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \left(y + y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 89.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -4200000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{-z}, a, x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - z}, a, x\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (<= z -4200000.0)
   (fma (/ (- z y) (- z)) a x)
   (if (<= z 2.9e+25) (fma (/ y (- -1.0 t)) a x) (fma (/ z (- t z)) a x))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if (z <= -4200000.0) {
		tmp = fma(((z - y) / -z), a, x);
	} else if (z <= 2.9e+25) {
		tmp = fma((y / (-1.0 - t)), a, x);
	} else {
		tmp = fma((z / (t - z)), a, x);
	}
	return tmp;
}

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if (z <= -4200000.0)
		tmp = fma(Float64(Float64(z - y) / Float64(-z)), a, x);
	elseif (z <= 2.9e+25)
		tmp = fma(Float64(y / Float64(-1.0 - t)), a, x);
	else
		tmp = fma(Float64(z / Float64(t - z)), a, x);
	end
	return tmp
end

Wolfram

code[x_, y_, z_, t_, a_] := If[LessEqual[z, -4200000.0], N[(N[(N[(z - y), $MachinePrecision] / (-z)), $MachinePrecision] * a + x), $MachinePrecision], If[LessEqual[z, 2.9e+25], N[(N[(y / N[(-1.0 - t), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision], N[(N[(z / N[(t - z), $MachinePrecision]), $MachinePrecision] * a + x), $MachinePrecision]]]

Derivation

1. Split input into 3 regimes

2. if z < -4.2e6

   1. Initial program 90.8%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around inf

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

      1. mul-1-neg N/A

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

      2. lower-neg.f64 81.2

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

   8. Applied rewrites 81.2%

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

   if -4.2e6 < z < 2.8999999999999999e25

   1. Initial program 99.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in z around 0

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

      1. associate-*r/ N/A

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

      2. lower-/.f64 N/A

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

      3. mul-1-neg N/A

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

      4. lower-neg.f64 N/A

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

      5. lift-+.f64 88.9

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

   8. Applied rewrites 88.9%

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

   if 2.8999999999999999e25 < z

   1. Initial program 94.1%

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

   3. Taylor expanded in a around 0

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

      1. +-commutative N/A

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

      2. *-commutative N/A

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

      3. lower-fma.f64 N/A

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

      4. sub-div N/A

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

      5. lower-/.f64 N/A

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

      6. lower--.f64 N/A

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

      7. lower--.f64 N/A

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

      8. lower-+.f64 99.9

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

   5. Applied rewrites 99.9%

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

   6. Taylor expanded in t around inf

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

   8. Applied rewrites 99.9%

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

   9. Taylor expanded in y around 0

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

   11. Applied rewrites 91.5%

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

4. Final simplification 88.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -4200000:\\ \;\;\;\;\mathsf{fma}\left(\frac{z - y}{-z}, a, x\right)\\ \mathbf{elif}\;z \leq 2.9 \cdot 10^{+25}:\\ \;\;\;\;\mathsf{fma}\left(\frac{y}{-1 - t}, a, x\right)\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(\frac{z}{t - z}, a, x\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 73.9% accurate, 1.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -1200000 \lor \neg \left(z \leq 2.8\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \left(y + y \cdot z\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -1200000.0) (not (<= z 2.8)))
   (- x a)
   (- x (* a (+ y (* y z))))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -1200000.0) || !(z <= 2.8)) {
		tmp = x - a;
	} else {
		tmp = x - (a * (y + (y * z)));
	}
	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 <= (-1200000.0d0)) .or. (.not. (z <= 2.8d0))) then
        tmp = x - a
    else
        tmp = x - (a * (y + (y * z)))
    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 <= -1200000.0) || !(z <= 2.8)) {
		tmp = x - a;
	} else {
		tmp = x - (a * (y + (y * z)));
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -1200000.0) or not (z <= 2.8):
		tmp = x - a
	else:
		tmp = x - (a * (y + (y * z)))
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -1200000.0) || !(z <= 2.8))
		tmp = Float64(x - a);
	else
		tmp = Float64(x - Float64(a * Float64(y + Float64(y * z))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -1200000.0) || ~((z <= 2.8)))
		tmp = x - a;
	else
		tmp = x - (a * (y + (y * z)));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -1200000.0], N[Not[LessEqual[z, 2.8]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[(x - N[(a * N[(y + N[(y * z), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -1.2e6 or 2.7999999999999998 < z

   1. Initial program 92.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 75.3%

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

   if -1.2e6 < z < 2.7999999999999998

   1. Initial program 99.1%

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

   3. Taylor expanded in t around 0

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

      1. lower-/.f64 N/A

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

      2. *-commutative N/A

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

      3. lower-*.f64 N/A

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

      4. lift--.f64 N/A

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

      5. lower--.f64 75.0

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

   5. Applied rewrites 75.0%

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

   6. Taylor expanded in y around inf

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lift--.f64 70.6

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

   8. Applied rewrites 70.6%

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

   9. Taylor expanded in z around 0

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

       1. lower-+.f64 N/A

          \[\leadsto x - a \cdot \left(y + y \cdot z\right) \]

       2. lower-*.f64 70.6

          \[\leadsto x - a \cdot \left(y + y \cdot z\right) \]

   11. Applied rewrites 70.6%

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

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -1200000 \lor \neg \left(z \leq 2.8\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot \left(y + y \cdot z\right)\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 73.9% accurate, 1.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -600000 \lor \neg \left(z \leq 2.8\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot y\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -600000.0) (not (<= z 2.8))) (- x a) (- x (* a y))))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -600000.0) || !(z <= 2.8)) {
		tmp = x - a;
	} else {
		tmp = x - (a * y);
	}
	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 <= (-600000.0d0)) .or. (.not. (z <= 2.8d0))) then
        tmp = x - a
    else
        tmp = x - (a * y)
    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 <= -600000.0) || !(z <= 2.8)) {
		tmp = x - a;
	} else {
		tmp = x - (a * y);
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -600000.0) or not (z <= 2.8):
		tmp = x - a
	else:
		tmp = x - (a * y)
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -600000.0) || !(z <= 2.8))
		tmp = Float64(x - a);
	else
		tmp = Float64(x - Float64(a * y));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -600000.0) || ~((z <= 2.8)))
		tmp = x - a;
	else
		tmp = x - (a * y);
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -600000.0], N[Not[LessEqual[z, 2.8]], $MachinePrecision]], N[(x - a), $MachinePrecision], N[(x - N[(a * y), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if z < -6e5 or 2.7999999999999998 < z

   1. Initial program 92.8%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 75.3%

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

   if -6e5 < z < 2.7999999999999998

   1. Initial program 99.1%

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

   3. Taylor expanded in z around 0

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

      1. associate-/l* N/A

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

      2. lower-*.f64 N/A

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

      3. lower-/.f64 N/A

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

      4. lower-+.f64 89.9

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

   5. Applied rewrites 89.9%

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

   6. Taylor expanded in t around 0

      \[\leadsto x - a \cdot y \]
   7. Step-by-step derivation

   8. Applied rewrites 70.6%

      \[\leadsto x - a \cdot y \]
3. Recombined 2 regimes into one program.

4. Final simplification 72.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -600000 \lor \neg \left(z \leq 2.8\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x - a \cdot y\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 66.7% accurate, 2.2× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;z \leq -470000 \lor \neg \left(z \leq 42000\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \end{array} \]

FPCore

(FPCore (x y z t a)
 :precision binary64
 (if (or (<= z -470000.0) (not (<= z 42000.0))) (- x a) x))

C

double code(double x, double y, double z, double t, double a) {
	double tmp;
	if ((z <= -470000.0) || !(z <= 42000.0)) {
		tmp = x - a;
	} else {
		tmp = x;
	}
	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 <= (-470000.0d0)) .or. (.not. (z <= 42000.0d0))) then
        tmp = x - a
    else
        tmp = x
    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 <= -470000.0) || !(z <= 42000.0)) {
		tmp = x - a;
	} else {
		tmp = x;
	}
	return tmp;
}

Python

def code(x, y, z, t, a):
	tmp = 0
	if (z <= -470000.0) or not (z <= 42000.0):
		tmp = x - a
	else:
		tmp = x
	return tmp

Julia

function code(x, y, z, t, a)
	tmp = 0.0
	if ((z <= -470000.0) || !(z <= 42000.0))
		tmp = Float64(x - a);
	else
		tmp = x;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x, y, z, t, a)
	tmp = 0.0;
	if ((z <= -470000.0) || ~((z <= 42000.0)))
		tmp = x - a;
	else
		tmp = x;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_, y_, z_, t_, a_] := If[Or[LessEqual[z, -470000.0], N[Not[LessEqual[z, 42000.0]], $MachinePrecision]], N[(x - a), $MachinePrecision], x]

Derivation

1. Split input into 2 regimes

2. if z < -4.7e5 or 42000 < z

   1. Initial program 92.7%

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

   3. Taylor expanded in z around inf

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

   5. Applied rewrites 76.2%

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

   if -4.7e5 < z < 42000

   1. Initial program 99.1%

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

   3. Taylor expanded in x around inf

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

   5. Applied rewrites 50.4%

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

4. Final simplification 62.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;z \leq -470000 \lor \neg \left(z \leq 42000\right):\\ \;\;\;\;x - a\\ \mathbf{else}:\\ \;\;\;\;x\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 54.1% accurate, 35.0× speedup

Math

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

FPCore

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

C

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

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
end function

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 96.1%

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

3. Taylor expanded in x around inf

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

5. Applied rewrites 50.3%

   \[\leadsto \color{blue}{x} \]
6. Add Preprocessing

Reproduce

herbie shell --seed 2025085 
(FPCore (x y z t a)
  :name "Graphics.Rendering.Chart.SparkLine:renderSparkLine from Chart-1.5.3"
  :precision binary64

  :alt
  (! :herbie-platform default (- x (* (/ (- y z) (+ (- t z) 1)) a)))

  (- x (/ (- y z) (/ (+ (- t z) 1.0) a))))